torsional pendulum experiment use

Microscale torsion resonators for short-range gravity experiments

Measuring gravitational interactions on sub-100- μ μ \upmu roman_μ m length scales offers a window into physics beyond the Standard Model. However, short-range gravity experiments are limited by the ability to position sufficiently massive objects to within small separation distances. Here we propose mass-loaded silicon nitride ribbons as a platform for testing the gravitational inverse square law at separations currently inaccessible with traditional torsion balances. These microscale torsion resonators benefit from low thermal noise due to strain-induced dissipation dilution while maintaining compact size ( < 100 ⁢ μ absent 100 μ <100~{}\upmu < 100 roman_μ g) to allow close approach. Considering an experiment combining a 40 ⁢ μ 40 μ 40~{}\upmu 40 roman_μ g torsion resonator with a source mass of comparable size ( 130 ⁢ μ 130 μ 130~{}\upmu 130 roman_μ g) at separations down to 25 μ μ ~{}\upmu roman_μ m, and including limits from thermomechanical noise and systematic uncertainty, we predict these devices can set novel constraints on Yukawa interactions within the 1-100 μ μ ~{}\upmu roman_μ m range.

Measurements of the gravitational inverse square law (ISL) at short distances can test physics beyond the Standard Model  [ 1 , 2 ] . Dark energy generically suggests a scale for new physics around 100 μ μ ~{}\upmu roman_μ m  [ 3 ] , and several specific theories have been proposed in this regime. For example, “fat gravitons” with size in the 20-95 μ μ ~{}\upmu roman_μ m range may solve the cosmological constant problem  [ 4 , 5 ] , chameleon fields can have interaction lengths as low as ∼ 25 ⁢ μ similar-to absent 25 μ {\sim 25}~{}\upmu ∼ 25 roman_μ m in a laboratory setting  [ 6 ] , and modifications to gravity from a “dark dimension” are predicted to arise between 1-10 μ μ ~{}\upmu roman_μ m  [ 7 , 8 ] . Experimental tests of the ISL are also sensitive to short-range Yukawa interactions mediated by massive particles such as dilatons  [ 9 ] , radions  [ 10 ] , or gauge bosons  [ 11 , 12 ] . In this context, one can parametrize a violation of the ISL with a Yukawa potential that augments the Newtonian gravitational potential as

Introduction A body's inherent resistance to rotational acceleration about a specific axis is called its' mass moment of inertia about that axis. Given a fixed applied torque, an object with a higher mass moment of inertia will have a lower rotational acceleration than a body with a lower moment of inertia (in accordance with the relationship T = J α ). This is the rotational equivalent to an object's mass being a measure of its inherent resistance to translational acceleration while a force is applied (in accordance with Newton's F = m a ). A rigorous definition of a moment of inertia can be found at Wolfram Research's Science World. The goals of this case study are: To develop a feel for what the mass moment of inertia of an object should be. To demonstrate the utility of using a torsional pendulum to determine mass moments of inertia for complex geometries. To experimentally determine the mass moment of inertia of a flat disk. To analytically determine the mass moment of inertia of a flat disk. To compare the experimental versus analytical values obtained for the flat disk. To determine the mass moment of inertia of a complex, concave link. While the mass moment of inertia can, in theory, be analytically determined for any object, doing so for more than simple shapes can quickly become difficult. Most Dynamics textbooks will contain an appendix detailing the mass moment of inertia of homogeneous simple 2D and 3D shapes (e.g. Meriam and Kraige's Engineering Mechanics: Dynamics , Fourth Edition). Determining values for superpositions of simple shapes requires using the Parallel Axis Theorem : If a body with mass m has a mass moment of inertia I about its mass center, then its mass moment of inertia about a different, parallel axis (denoted J ¯ ) a distance d from the mass center is given by the expression J ¯ = J + m d 2 . The first axis must be the mass center, and the two axes must be parallel. A torsional pendulum is a device which can be used to experimentally measure mass moments of inertia for arbitrarily shaped objects. The pendulum is essentially a vertically-mounted torsional spring where the top side is fixed. The body whose mass moment is to be measured is suspended from the bottom side of the spring. The object is then rotated slightly and released so that small rotary oscillations occur. Given the period of the oscillation and a little bit of math, the mass moment of inertia can be experimentally calculated. Wolfram's Science World also has a brief explanation of the torsional pendulum . A description of the procedure for this experiment is as follows: This experiment will require: A torsional pendulum. This one consists of an upper platform to which three strings are attached. The three strings (a little difficult to see) attach to the flat disk located at the bottom of the images. The strings act as the torsional spring in this apparatus. A flat disk (seen also in the pendulum picture). Its mass moment of inertia will be found both experimentally and analytically in the first part of the experiment. A complex, concave link taken from a four bar web cutter mechanism taken from Haug (see references). Its mass moment of inertia will be found experimentally in the second part of the experiment. Also: A stopwatch, a mass balance or scale, a ruler, a level, an small edge on which to balance the concave link (more detail on this in the procedure), and a flat table surface. Initial Setup Measure both the outer radius of the circular disk and the radius from its center to the attachment of the strings (they are not the same for this setup). Determine the mass of the disk. This disk will serve as the lower platform for the torsional pendulum. The pendulum should be placed on the flat table surface. The upper disk should be checked to be sure it is level. The suspended circular platform should then be checked for levelness. If it is not level while at rest, the lengths of the three strings should be adjusted until the lower platform is level. The pendulum is now ready for use. Procedure for Determining Moment of Inertia for the Disk The lower platform should be started into small rotational oscillation. It is important that the oscillations be small, and that translational motion of the platform be minimized as much as possible; this is necessary to obtain good results. Using the stopwatch, measure the time necessary for the the platform to complete twenty oscillatory periods. Repeat ten times. The large number of samples is important to obtain a good measure of the mean oscillation period. Procedure for Determining Moment of Inertia for Concave Linkage Determine the mass of the complex, concave link. It is important that the concave link be placed so that its mass center lies at the center of the lower circular platform. Since the link is irregularly shaped it is hard to find its centroid by eye. One way to determine the mass center is to balance the link on some sort of raised edge. Mark the line along which the edge balances. Then balance the link in a different orientation. Mark the second line along which the link balances. The mass center of the link will lie at the intersection of these two lines. Note that for this concave link the mass center does not physically lie on the link itself; it lies inside the concavity. To the left is a picture of the link being balanced. Place the link with its centroid at the center of the lower platform to ensure that their centroid axes coincide. As before, start the torsional pendulum so that it oscillates in a small, rotary motion. Using the stopwatch, measure the time for the platform and link to complete twenty torsional periods. Repeat ten times. Download Files File Title Type Size Video of Torsional Oscillation MPEG Movie 1.3M Geometry and Mass Information Length of pendulum strings 23.0625 inches Outer radius of the disk 3 inches Radius to string attachments 2.75 inches Mass of the circular disk 152.97 grams Mass of the concave link 149.57 grams Oscillation Time Data Times (in seconds) for the circular lower platform to complete twenty torsional oscillations: 23.87 23.68 24.28 23.59 23.87 23.85 23.72 23.56 23.94 23.81 Times (in seconds) for the circular lower platform and the concave link to complete twenty torsional oscillations: 25.66 26.09 25.86 25.86 25.81 25.81 25.82 25.81 25.87 26.15 References used in creating this case study: Meriam and Kraige. Engineering Mechanics: Dynamics . Fourth Edition. John Wiley and Sons. 1997. "Eric Weisstein's World of Physics: Torsional Pendulum" from http://scienceworld.wolfram.com/physics/TorsionalPendulum.html Courtesy of Wolfram Research Inc. "Eric Weisstein's World of Physics: Moment of Inertia" from http://scienceworld.wolfram.com/physics/MomentOfInertia.html Courtesy of Wolfram Research Inc. Web Cutter Mechanism image scanned by Professor H. J. Sommer from Edward Haug. Computer Aided Kinematics and Dynamics of Mechanical Systems, Volume 1 . Allyn and Bacon. 1989. J. M. Cimbala. ME 82 - Mechanical Engineering Measurements course web pages at http://www.me.psu.edu/me82/ Thanks to Professor Sommer for his help in putting together this case study, and for access to his previous assignments regarding this experiment. Calculations for the Moment of Inertia for the Disk The time necessary for one oscillation of the circular disk should be calculated for each of the ten sampling periods. Calculate the mean and standard deviation of a single oscillation period. Once this is found, the experimental mass moment of inertia of the circular lower platform can be found from: gmr2τ2 4sπ2 J is the mass moment of inertia of the object(s) g is the acceleration due to gravity m is the mass of the object(s) r is the radius from the center of the disk to the attachment point of the strings ( not the outer radius of the disk) τ is the time period for one torsional oscillation s is the length of the string from the upper platform to the top of the disk Determine the analytical mass moment of inertia for the disk about its center based on its mass and its outer radius. The formula is readily available in most dynamics textbooks. Calculations for the Moment of Inertia for the Concave Link The calculations for determining the mass moment of inertia for the complex, concave link are similar to the ones for finding the value for the circular disk alone. Two changes are necessary. The first is that now J = J platform + J link since mass moments obey superposition principles. Use the experimental J link found above for this expression. The second is that the mass m in the earlier equation is now m = m platform + m link since both bodies are undergoing rotation. Statistical Analysis Calculate the mean and standard deviation for one oscillation for both the disk alone and the disk/link combination. Find the experimental and analytical mass moment of inertia of the disk. Find and compare the error between the two. Does the experimental value match the analytical? Using Single Point Expected RSS Uncertainty Analysis (see Dr. Cimbala's ME 82 statistical analysis lecture notes ), report the expected uncertainty of the experimental mass moment of inertia measurement from the oscillation period standard deviation. Calculate the experimental mass moment of inertia of the concave link. Does it seem reasonable? Optional Analytically determine the mass moment of the link (using the picture found in the equipment section and the ruler placed in that image for scale). How closely does the analytical value match the experimental? If the link was placed so that its centroid was a quarter of an inch away from the centroid of the platform, how much would the oscillation period of the platform/link combination change? (hint: use the parallel axis theorem with d = 0.25 inches to find the corresponding change in J link ). How could you measure the mass moment of inertia of an automobile about all three axes through its mass centroid (i.e. vertical, horizontal right/left, horizontal front/back)? Torsion Pendulum In torsion pendulums (like the one used in some clocks), the oscillation speed depends on the moment of inertia of the pendulum system and the spring constant of the wire, which is a measure of how torque is required to twist the wire through some angle. (See the Angular Momentum Turntable exhibit for more on the moment of inertia.) Experiment 1: WHAT TO DO: Place the two large masses on the outer two pins of the torsion arm. Pull one side of the torsion arm towards you a small distance and then let it go. Count the time it takes the arm to make ten oscillations. Replace the large masses with the two small masses in the same spots. Pull one side of the torsion arm towards you the same small distance as before and let it go. HOW DO THE OSCILLATION TIMES (PERIODS) COMPARE? Answer: The period of the torsion arm with the larger masses is twice as long as the period of the arm with the smaller masses. Experiment 2: WHAT TO DO: Put the small masses on the same outer two pins of the torsion arm. Remove the small masses. Place the two large masses on the INNER two pins of the torsion arm. Answer: The period of the torsion arm with the smaller masses is equal to the period of the torsion arm with the larger masses. What is going on? The moment of inertia, I , depends on the mass on the pendulum and its distance from the center of rotation of the pendulum. A smaller the moment of inertia increases the oscillation frequency and decreases the period of the pendulum. In Exp.1, the pendulum has a higher I with the larger masses than with the small, so the period is longer than with the small masses. In Exp. 2, the moment of inertia for the small masses at a big distance is equal to the moment of inertia for the big masses at a small distance, so the periods are the same in both cases. Original 1918 Museum Exhibit Physics Lecture Demonstration Database Torsion Pendulum, 3A10.30 PHYSICS2CHEMISTRY Exploring torsional pendulum: properties, uses, and applications, torsional pendulum. A torsional pendulum is a simple harmonic oscillator that consists of a disk or a sphere suspended from a thin wire or a fiber, which is twisted and then released to oscillate back and forth. The twisting of the wire provides a restoring torque that acts on the pendulum, causing it to oscillate with a torsional frequency. Here are some characteristics and properties of a torsional pendulum : ⇒  Period : The period of oscillation of a torsional pendulum is given by T=2π√(I/κ), where I is the moment of inertia of the pendulum, and κ is the torsional constant of the wire. The period of oscillation is independent of the amplitude of the oscillation, and depends only on the properties of the pendulum. ⇒  Torsional constant : The torsional constant κ of the wire or fiber is a measure of its stiffness, and determines the magnitude of the restoring torque acting on the pendulum. It is proportional to the length of the wire, and inversely proportional to its diameter and the modulus of elasticity of the material. ⇒  Damping : Like any oscillating system, a torsional pendulum is subject to damping due to friction or other dissipative forces. The damping of the pendulum can be characterized by its quality factor Q, which is a measure of the ratio of the energy stored in the pendulum to the energy lost per cycle of oscillation. ⇒  Resonance : A torsional pendulum can exhibit resonance when it is subjected to an external force at its natural frequency. At resonance, the amplitude of the oscillation can be greatly increased, and the energy transfer between the pendulum and the external force is maximized. ⇒  Applications : Torsional pendulums are commonly used in experimental physics to measure the torsional constants of wires and fibers, and to study the properties of materials such as viscoelastic fluids and polymers. They are also used in precision timekeeping, such as in atomic clocks and gyroscopes. Frequently Asked Questions – FAQs ⇒  What is the difference between a torsional pendulum and a simple pendulum? A torsional pendulum oscillates back and forth in a rotational motion, while a simple pendulum oscillates back and forth in a linear motion. A torsional pendulum is also subject to torsional restoring forces, while a simple pendulum is subject to gravitational restoring forces. ⇒  What is the period of a torsional pendulum? The period of a torsional pendulum depends on the torsional constant of the wire or fiber and the moment of inertia of the pendulum. The period can be calculated using the equation T=2π√(I/κ), where T is the period, I is the moment of inertia, and κ is the torsional constant. ⇒  How is the torsional constant of a wire or fiber measured? The torsional constant of a wire or fiber can be measured using a torsional pendulum by measuring the period of oscillation and using the equation κ=(4π²I)/T², where T is the period and I is the moment of inertia of the pendulum. ⇒  What is the quality factor of a torsional pendulum? The quality factor of a torsional pendulum is a measure of the damping in the system, and is defined as the ratio of the energy stored in the pendulum to the energy lost per cycle of oscillation. A high quality factor indicates low damping and a longer decay time. ⇒  How does damping affect the oscillation of a torsional pendulum? Damping in a torsional pendulum reduces the amplitude of oscillation and causes the oscillation to decay over time. The rate of decay depends on the quality factor of the pendulum and the amount of damping present. ⇒  What is resonance in a torsional pendulum? Resonance in a torsional pendulum occurs when an external force is applied at the natural frequency of the pendulum, causing the amplitude of oscillation to increase. The amplitude can be greatly increased at resonance, leading to larger oscillations and more efficient energy transfer. ⇒  What are some applications of torsional pendulums? Torsional pendulums are used in a variety of applications, including measuring the torsional constants of wires and fibers, studying the properties of viscoelastic materials, and as components in atomic clocks and gyroscopes. ⇒  How can the moment of inertia of a pendulum be calculated? The moment of inertia of a pendulum can be calculated using the equation I=mr², where m is the mass of the pendulum and r is the radius of gyration. The radius of gyration is a measure of how the mass of the pendulum is distributed around the axis of rotation. Let me know if you have more questions or if there is a specific topic that you would like to know more about. You may like these posts Post a comment. If you have any doubts, please let me know Top Post Ad Below post ad, footer copyright, report abuse, search this blog. April 2023 17 March 2023 62 February 2023 40 January 2023 62 December 2022 45 October 2022 5 September 2022 4 July 2022 2 June 2022 11 April 2022 18 July 2021 3 June 2021 12 Chemistry Experiments condense matter Difference Between Electricity and Magnetism Electronics Inorganic Chemistry Light and optics Organic Chemistry Physical Chemistry Physics Experiments Solid State Thermodynamics Waves and Vibrations Popular Posts What are lanthanoids and actinoids?| difference between lanthanides and actinides| discovered | USES | general formula| Determine the ionization constant of a weak acid conductometrically. Separate the leaf pigments from spinach leaves by column chromatography. Chemistry Experiments 30 condense matter 13 Difference Between 111 Electricity and Magnetism 21 Electronics 47 Inorganic Chemistry 47 Light and optics 20 Mechanics 20 Organic Chemistry 45 Physical Chemistry 58 Physics Experiments 6 Solid State 4 Thermodynamics 10 Waves and Vibrations 12 Privacy Policy Terms of Use Copyright (c) 2023 Physics2Chemistry All Right Reseved Contact Form Teacher Resource Center Pasco partnerships. 2024 Catalogs & Brochures Rotation and torque experiments. This set of rotational and torque experiments can be performed with the equipment from the Rotational Motion Plus Kit (ME-1261). Grade Level: College • High School Subject: Physics 02) Torsional Pendulum The torsional pendulum consists of a torsion wire attached to a Rotary Motion Sensor with an object (a disk, a ring, or a rod with point masses) mounted on top of it. The period of oscillation is measured from a plot of the angular displacement versus time. To calculate the theoretical period, the rotational inertia is determined by measuring the dimensions of the object and the torsional spring constant is determined from the slope of a plot of force versus angular displacement. 03) Gravitational Torque Students set up various systems to learn about gravitational torque and center of mass. They find the mass of an object, determine the mass of a meter stick, predict the location of a mass to balance an off-center meter stick, and locate the center of mass of an irregular object. 04) Exploring a Rotating System Students construct and collect data with an experimental system to determine angular velocity, angular acceleration, applied torque, and the rotational inertia of the meter stick component. 05) Exploring Physical Pendulums Students use a meter stick as a physical pendulum to explore the factors that affect the period and the mathematical properties of the physical pendulum period equation. 06) Centripetal, Tangential, and Angular Acceleration A rod rotates in a horizontal plane, and is made to slow steadily to a stop. This setup is used to explore the different types of acceleration involved in this motion: centripetal, tangential, and angular acceleration. 07) Rotational Inertia The purpose of this experiment is to find the rotational inertia of a ring and a disk experimentally and to verify that these values correspond to the calculated theoretical values. 08) Newton’s Second Law for Rotation Newton's Second Law for rotation: The resulting angular acceleration (α) of an object is directly proportional to the net torque (τ) on that object. The hanging mass applies a torque to the shaft of the Rotary Motion Sensor and the resulting angular acceleration of the rod and brass masses is investigated. 09) Rotational Kinetic Energy This lab investigates the potential energies for a modified Atwood's Machine, where a disk has been added to the Rotary Motion Sensor pulley. 10) Conservation of Angular Momentum A non-rotating ring is dropped onto a rotating disk. The angular speed is measured immediately before the drop and after the ring stops sliding on the disk. The measurements are repeated with a non-rotating disk being dropped onto a rotating disk. For each situation, the initial angular momentum is compared to the final angular momentum. Initial and final kinetic energy are also calculated and compared. 11) Conservation of Energy of a Simple Pendulum The purpose of this experiment is to use measurements of the motion of a simple pendulum to calculate and compare the different types of energy present in the system. 12) Physical Pendulum A rod oscillates as a physical pendulum. The period is measured directly by the Rotary Motion Sensor, and the value is compared to the theoretical period calculated from the dimensions of the pendulum. 13) Large Amplitude Pendulum This experiment explores the oscillatory motion of a physical pendulum for both small and large amplitudes. Waveforms are examined for angular displacement, velocity and acceleration, and the dependence of the period of a pendulum on the amplitude of oscillation is investigated. Related Products Account Required Downloading files for this experiment requires a PASCO account. Experiment Features Teacher files, sparkvue files, pasco capstone files, wireless sensors, pasport sensors, scienceworkshop sensors. Many lab activities can be conducted with our Wireless , PASPORT , or even ScienceWorkshop sensors and equipment. For assistance with substituting compatible instruments, contact PASCO Technical Support . We're here to help. Torsional Pendulum A torsional pendulum consists of a disk (or some other object) suspended from a wire, which is then twisted and released, resulting in an oscillatory motion.  The oscillatory motion is caused by a restoring torque which is proportional to the angular displacement, Similar to the simple pendulum, so long as the angular displacement is small (which means the motion is SHM) the period is independent of the displacement.  Torsional pendulums are also used as a time keeping devices , as in for example, the mechanical wristwatch . Copernicus' parents: Copernicus, young man, when are you going to come to terms with the fact that the world does not revolve around you?! Dr. C. L. Davis Physics Department University of Louisville email : [email protected]   15.4 Pendulums Learning objectives. By the end of this section, you will be able to: State the forces that act on a simple pendulum Determine the angular frequency, frequency, and period of a simple pendulum in terms of the length of the pendulum and the acceleration due to gravity Define the period for a physical pendulum Define the period for a torsional pendulum Pendulums are in common usage. Grandfather clocks use a pendulum to keep time and a pendulum can be used to measure the acceleration due to gravity. For small displacements, a pendulum is a simple harmonic oscillator. The Simple Pendulum A simple pendulum is defined to have a point mass, also known as the pendulum bob , which is suspended from a string of length L with negligible mass ( Figure 15.20 ). Here, the only forces acting on the bob are the force of gravity (i.e., the weight of the bob) and tension from the string. The mass of the string is assumed to be negligible as compared to the mass of the bob. Consider the torque on the pendulum. The force providing the restoring torque is the component of the weight of the pendulum bob that acts along the arc length. The torque is the length of the string L times the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement: The solution to this differential equation involves advanced calculus, and is beyond the scope of this text. But note that for small angles (less than 15 degrees or about 0.26 radians), sin θ sin θ and θ θ differ by less than 1%, if θ is measured in radians. We can then use the small angle approximation sin θ ≈ θ . sin θ ≈ θ . The angle θ θ describes the position of the pendulum. Using the small angle approximation gives an approximate solution for small angles, Because this equation has the same form as the equation for SHM, the solution is easy to find. The angular frequency is and the period is The period of a simple pendulum depends on its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass and the maximum displacement. As with simple harmonic oscillators, the period T for a pendulum is nearly independent of amplitude, especially if θ θ is less than about 15 ° . 15 ° . Even simple pendulum clocks can be finely adjusted and remain accurate. Note the dependence of T on g . If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity, as in the following example. Example 15.3 Measuring acceleration due to gravity by the period of a pendulum. Square T = 2 π L g T = 2 π L g and solve for g : g = 4 π 2 L T 2 . g = 4 π 2 L T 2 . Substitute known values into the new equation: g = 4 π 2 0.75000 m ( 1.7357 s ) 2 . g = 4 π 2 0.75000 m ( 1.7357 s ) 2 . Calculate to find g : g = 9.8281 m/s 2 . g = 9.8281 m/s 2 . Significance Check your understanding 15.4. An engineer builds two simple pendulums. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10 kg. Pendulum 2 has a bob with a mass of 100 kg. Describe how the motion of the pendulums will differ if the bobs are both displaced by 12 ° 12 ° . Physical Pendulum Any object can oscillate like a pendulum. Consider a coffee mug hanging on a hook in the pantry. If the mug gets knocked, it oscillates back and forth like a pendulum until the oscillations die out. We have described a simple pendulum as a point mass and a string. A physical pendulum is any object whose oscillations are similar to those of the simple pendulum, but cannot be modeled as a point mass on a string, and the mass distribution must be included into the equation of motion. As for the simple pendulum, the restoring force of the physical pendulum is the force of gravity. With the simple pendulum, the force of gravity acts on the center of the pendulum bob. In the case of the physical pendulum, the force of gravity acts on the center of mass (CM) of an object. The object oscillates about a point O . Consider an object of a generic shape as shown in Figure 15.21 . When a physical pendulum is hanging from a point but is free to rotate, it rotates because of the torque applied at the CM, produced by the component of the object’s weight that acts tangent to the motion of the CM. Taking the counterclockwise direction to be positive, the component of the gravitational force that acts tangent to the motion is − m g sin θ − m g sin θ . The minus sign is the result of the restoring force acting in the opposite direction of the increasing angle. Recall that the torque is equal to τ → = r → × F → τ → = r → × F → . The magnitude of the torque is equal to the length of the radius arm times the tangential component of the force applied, | τ | = r F sin θ | τ | = r F sin θ . Here, the length L of the radius arm is the distance between the point of rotation and the CM. To analyze the motion, start with the net torque. Like the simple pendulum, consider only small angles so that sin θ ≈ θ sin θ ≈ θ . Recall from Fixed-Axis Rotation on rotation that the net torque is equal to the moment of inertia I = ∫ r 2 d m I = ∫ r 2 d m times the angular acceleration α , α , where α = d 2 θ d t 2 α = d 2 θ d t 2 : Using the small angle approximation and rearranging: Once again, the equation says that the second time derivative of the position (in this case, the angle) equals minus a constant ( − m g L I ) ( − m g L I ) times the position. The solution is where Θ Θ is the maximum angular displacement. The angular frequency is The period is therefore Note that for a simple pendulum, the moment of inertia is I = ∫ r 2 d m = m L 2 I = ∫ r 2 d m = m L 2 and the period reduces to T = 2 π L g T = 2 π L g . Example 15.4 Reducing the swaying of a skyscraper. Find the moment of inertia for the CM: Use the parallel axis theorem to find the moment of inertia about the point of rotation for a rod length L : I = I CM + L 4 2 M = 1 12 M L 2 + 1 4 M L 2 = 1 3 M L 2 . I = I CM + L 4 2 M = 1 12 M L 2 + 1 4 M L 2 = 1 3 M L 2 . The period of a physical pendulum with its center of mass a distance l from the pivot point has a period of T = 2 π I m g l T = 2 π I m g l . In this problem, L has been defined as the total length of the rod, so l = L / 2 l = L / 2 . Use the moment of inertia to solve for the length L : T = 1 3 M L 2 M g L 2 = 2 π 2 L 3 g ; L = 3 T 2 g 8 π 2 = 1.49 m . T = 1 3 M L 2 M g L 2 = 2 π 2 L 3 g ; L = 3 T 2 g 8 π 2 = 1.49 m . Torsional Pendulum A torsional pendulum consists of a rigid body suspended by a light wire or spring ( Figure 15.22 ). When the body is twisted some small maximum angle ( Θ ) ( Θ ) and released from rest, the body oscillates between ( θ = + Θ ) ( θ = + Θ ) and ( θ = − Θ ) ( θ = − Θ ) . The restoring torque is supplied by the shearing of the string or wire. The restoring torque can be modeled as being proportional to the angle: The variable kappa ( κ ) ( κ ) is known as the torsion constant of the wire or string. The minus sign shows that the restoring torque acts in the opposite direction to increasing angular displacement. The net torque is equal to the moment of inertia times the angular acceleration: This equation says that the second time derivative of the position (in this case, the angle) equals a negative constant times the position. This looks very similar to the equation of motion for the SHM d 2 x d t 2 = − k m x d 2 x d t 2 = − k m x , where the period was found to be T = 2 π m k T = 2 π m k . Therefore, the period of the torsional pendulum can be found using The units for the torsion constant are [ κ ] = N-m = ( kg m s 2 ) m = kg m 2 s 2 [ κ ] = N-m = ( kg m s 2 ) m = kg m 2 s 2 and the units for the moment of inertial are [ I ] = kg-m 2 , [ I ] = kg-m 2 , which show that the unit for the period is the second. Example 15.5 Measuring the torsion constant of a string. Find the moment of inertia for the CM: I CM = ∫ x 2 d m = ∫ − L / 2 + L / 2 x 2 λ d x = λ [ x 3 3 ] − L / 2 + L / 2 = λ 2 L 3 24 = ( M L ) 2 L 3 24 = 1 12 M L 2 . I CM = ∫ x 2 d m = ∫ − L / 2 + L / 2 x 2 λ d x = λ [ x 3 3 ] − L / 2 + L / 2 = λ 2 L 3 24 = ( M L ) 2 L 3 24 = 1 12 M L 2 . Calculate the torsion constant using the equation for the period: T = 2 π I κ ; κ = I ( 2 π T ) 2 = ( 1 12 M L 2 ) ( 2 π T ) 2 ; = ( 1 12 ( 4.00 kg ) ( 0.30 m ) 2 ) ( 2 π 0.50 s ) 2 = 4.73 N · m . T = 2 π I κ ; κ = I ( 2 π T ) 2 = ( 1 12 M L 2 ) ( 2 π T ) 2 ; = ( 1 12 ( 4.00 kg ) ( 0.30 m ) 2 ) ( 2 π 0.50 s ) 2 = 4.73 N · m . This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax. Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction Authors: William Moebs, Samuel J. Ling, Jeff Sanny Publisher/website: OpenStax Book title: University Physics Volume 1 Publication date: Sep 19, 2016 Location: Houston, Texas Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction Section URL: https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums © Jul 23, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University. Office Hours Monday - Friday 8:00 AM - 5:00 PM (usually closed for lunch 12-1) For assistance, please contact us by email [email protected] or by calling 360-650-3818   Torsion Pendulum - Moment of Inertia Physics demonstrations. Demonstration Catalog Reservation Form Reserved Demonstrations Demonstration Equipment Preform the Torsion Pendulum experiment.  Disk or disk-with-ring or disk-with-two-cylinders can be chosen and a rod of brass of 0.865 m length and three different radii. Place the two assembly blocks on the torsion pendulum support board base and place the disk (and also ring if used) on top of them.  Pass the desired rod through the disk hole and then up into the rod hanger.  Tighter the thumb screw on the rod hanger and then the thumb screw on the disk.  Make sure the thumb screws are tightening into the rod end notches. Pull ot the two assembly blocks. The components all have labels with dimensions, masses, and moments of inertia for the components and the torsion constants for the rods. The linked document below has photos; complete dimensions, masses, and moments of inertia for the components; the torsion constants for the rods; equations; sample calculated periods; and sample measured periods. You can do as much or as little in lecture, using the calculations in the document. More Information Required equipment, related course. Faculty Resource Center Biochemistry Bioengineering Cancer Research Developmental Biology Engineering Environment Immunology and Infection Neuroscience JoVE Journal JoVE Encyclopedia of Experiments JoVE Chrome Extension Environmental Sciences Pharmacology JoVE Science Education JoVE Lab Manual JoVE Business Videos Mapped to your Course High Schools Videos Mapped to Your Course Chapter 15: Oscillations Back to chapter, torsional pendulum, previous video 15.7: problem solving: energy in simple harmonic motion, next video 15.12: damped oscillations. A torsional pendulum is a rigid body, like a top, suspended from a string that is assumed to be massless — an assumption that is valid if the rigid body's mass is much larger than the string's mass. When the top is twisted about the string's axis and released, it oscillates between two angles. The restoring torque is due to shearing of the string. If the angular displacement is small, the restoring torque can be modeled as proportional to the angular displacement. The proportionality constant is called the string's torsion constant. The torque can also be written in terms of the rigid body's moment of inertia and angular acceleration. The two expressions give an equation for simple harmonic motion, with the independent variable being the angle of oscillation, the mass replaced by the moment of inertia, and the force constant replaced by the string's torsion constant. The angular frequency of the oscillation is then determined, and from it, the time period is derived. A torsional pendulum involves the oscillation of a rigid body in which the restoring force is provided by the torsion in the string from which the rigid body is suspended. Ideally, the string should be massless; practically, its mass is much smaller than the rigid body's mass and is neglected. As long as the rigid body's angular displacement is small, its oscillation can be modeled as a linear angular oscillation. The amplitude of the oscillation is an angle. The role of mass is played by the rigid body's moment of inertia about the point of suspension and the axis passing perpendicular to it. Using the relationship between torque and angular acceleration, the equation is seen to mimic the equation of the simple harmonic motion of a simple pendulum. This observation allows for the easy determination of the angular frequency of the angular oscillation and its time period. Suggested Reading Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics . San Francisco, CA: Pearson: section 14.4; page 451. OpenStax. (2019). University Physics Vol. 1 . [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction : section 15.4; pages 769–770.      . For example, suspending a from a thin wire and winding it by an angle a . Therefore, the equation of motion is is the . But this is just a with equation of motion , ,

at a distance r 𝑟 r italic_r from a point-mass source M 𝑀 M italic_M . Here α 𝛼 \alpha italic_α is the interaction strength relative to Newtonian gravity and λ = ℏ / ( m b ⁢ c ) 𝜆 Planck-constant-over-2-pi subscript 𝑚 b 𝑐 \lambda=\hbar/(m_{\text{b}}c) italic_λ = roman_ℏ / ( italic_m start_POSTSUBSCRIPT b end_POSTSUBSCRIPT italic_c ) is the interaction length, where m b subscript 𝑚 b m_{\text{b}} italic_m start_POSTSUBSCRIPT b end_POSTSUBSCRIPT is the mass of the exchange boson.

A variety of devices have been used to search for Yukawa interactions in the sub-100- μ μ \upmu roman_μ m range, including torsion balances  [ 13 , 14 , 3 , 15 ] , optomechanical cantilevers  [ 16 ] , microelectromechanical torsion oscillators  [ 17 ] , levitated particles  [ 18 ] , and Casimir experiments  [ 19 , 20 ] . Of these, only cm-scale torsion balances with test masses exceeding 100 mg have achieved sensitivity to gravitational-strength ( | α | = 1 𝛼 1 \left|\alpha\right|=1 | italic_α | = 1 ) interactions  [ 3 , 21 , 22 , 23 , 15 , 24 ] . The upper bound for the interaction length of gravitational-strength Yukawa interactions has been progressively constrained by the Eöt-Wash group from 197 ⁢ μ 197 μ 197~{}\upmu 197 roman_μ m in 2001  [ 13 , 14 ] to 56 ⁢ μ 56 μ 56~{}\upmu 56 roman_μ m in 2007  [ 3 ] , to 42 μ μ ~{}\upmu roman_μ m in 2013  [ 22 ] , and most recently to 39 μ μ ~{}\upmu roman_μ m in 2020  [ 15 ] . Device planarity and the onset of electrostatic noise set the lower limit on the minimum surface separation between the source and test masses in torsion balance experiments  [ 15 , 24 ] , making it difficult to probe shorter interaction lengths.

Here we propose microscale torsion resonators as a platform for measuring gravity on sub-100- μ μ \upmu roman_μ m length scales. Introduced in Ref.  [ 26 ] , these devices are based on strained Si 3 N 4 nanoribbons that have been mass-loaded with a silicon pad to form sub-milligram, chip-scale torsion resonators. The compactness and optically flat planar geometry of the microresonators make them well suited for close approach to a planar source mass, with most of their mass able to participate in the short-range interaction due to the thinness of the pad. Furthermore, the optical lever readout of their motion does not require additional, non-participating mass, as the laser beam reflects directly off of the pad surface  [ 26 ] . With mechanical quality factors exceeding one million due to dissipation dilution in the strained Si 3 N 4 suspension, the devices reported in [ 26 ] possess room temperature thermal torque noise below 10 − 19 ⁢ N ⁢ m / Hz superscript 10 19 N m Hz 10^{-19}~{}\text{N}\,\text{m}/\sqrt{\text{Hz}} 10 start_POSTSUPERSCRIPT - 19 end_POSTSUPERSCRIPT N m / square-root start_ARG Hz end_ARG , with the potential for further noise reduction through cryogenic cooling  [ 16 ] . The devices may also be less susceptible to noise arising from electrostatic effects at narrow separations, as strain-induced stiffness minimizes the overall resonator motion  [ 27 ] . Finally, lithographically defined geometry ensures precise dimensional control, improving fabrication tolerances while also enabling rapid production of multiple units to gather statistics on fabrication errors.

An example of the proposed experiment is depicted in Fig. 1 , where the gravitational coupling between a torsion microresonator and a heterogeneous source mass of comparable size is measured as a function of separation s 𝑠 s italic_s . The source mass is continually rotated such that the gravitational torque on the pad oscillates at three times the rotation frequency (neglecting higher-order harmonics). Torque measurements at various separations can be used to distinguish a potential Yukawa interaction from a purely Newtonian gravitational signal, which would have a different s 𝑠 s italic_s -dependence, as depicted in Fig. 2 a. This procedure also helps discern contributions from separation-independent effects, such as wobble in the rotary system that produces vibrations at rotation frequency harmonics. To reduce electrostatic coupling between the source and test masses, an electrostatic shield is inserted between the two (omitted from Fig. 1 ), such as a metallized Si 3 N 4 membrane  [ 16 ] . However, coupling between the resonator and the shield can still have effects such as altering the resonator stiffness or presenting a noise source at close separations  [ 28 , 15 , 27 ] , potentially requiring the pad be metallized and grounded through partial metallization of the ribbon suspension  [ 29 ] . Remaining contact potentials can be compensated by applying a bias voltage to the shield  [ 24 ] .

The experiment’s sensitivity is fundamentally limited by thermal motion of the torsion resonator with a torque-equivalent power spectral density S τ th = 8 ⁢ π ⁢ k B ⁢ T ⁢ f 0 ⁢ I 0 / Q 0 superscript subscript 𝑆 𝜏 th 8 𝜋 subscript 𝑘 B 𝑇 subscript 𝑓 0 subscript 𝐼 0 subscript 𝑄 0 S_{\tau}^{\text{th}}=8\pi k_{\text{B}}Tf_{0}I_{0}/Q_{0} italic_S start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT = 8 italic_π italic_k start_POSTSUBSCRIPT B end_POSTSUBSCRIPT italic_T italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . While the moment of inertia I 0 = 3 × 10 − 15 subscript 𝐼 0 3 superscript 10 15 I_{0}=3\times 10^{-15} italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 3 × 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT kg m 2 only depends on the pad dimensions, the resonance frequency f 0 subscript 𝑓 0 f_{0} italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and quality factor Q 0 subscript 𝑄 0 Q_{0} italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT depend also on the ribbon suspension geometry (see Appendix B ). The ribbon is assumed to have a 150 ⁢ μ 150 μ 150~{}\upmu 150 roman_μ m width, 40 40 40~{} 40 nm thickness, and 1.5 cm total length. Following Ref. [ 26 ] , these parameters are expected to yield f 0 = 137 subscript 𝑓 0 137 f_{0}=137 italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 137 Hz, Q 0 = 6 × 10 7 subscript 𝑄 0 6 superscript 10 7 Q_{0}=6\times 10^{7} italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 6 × 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT , and S τ th = 3 × 10 − 20 superscript subscript 𝑆 𝜏 th 3 superscript 10 20 \sqrt{S_{\tau}^{\text{th}}}=3\times 10^{-20}~{} square-root start_ARG italic_S start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT end_ARG = 3 × 10 start_POSTSUPERSCRIPT - 20 end_POSTSUPERSCRIPT N m/ Hz Hz \sqrt{\rm Hz} square-root start_ARG roman_Hz end_ARG at T = 300 𝑇 300 T=300~{} italic_T = 300 K. Each torque measurement is polluted by thermal noise with standard deviation δ ⁢ N = S τ th / t meas 𝛿 𝑁 superscript subscript 𝑆 𝜏 th subscript 𝑡 meas \delta N=\sqrt{S_{\tau}^{\text{th}}/t_{\text{meas}}} italic_δ italic_N = square-root start_ARG italic_S start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT / italic_t start_POSTSUBSCRIPT meas end_POSTSUBSCRIPT end_ARG , depicted by dashed lines in Fig. 2 a.

A nonzero Yukawa interaction would manifest as a deviation from the expected gravitational torque, so the minimum detectable interaction strength α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT is limited by uncertainty in both the total torque estimate and the Newtonian torque model. In particular, errors in calibration, fabrication, or alignment may lead to imperfect subtraction of the expected Newtonian contribution. To account for these effects, consider a model for the torque on the paddle with contributions from Newtonian gravity τ G subscript 𝜏 G \tau_{\text{G}} italic_τ start_POSTSUBSCRIPT G end_POSTSUBSCRIPT and a Yukawa interaction τ Y subscript 𝜏 Y \tau_{\text{Y}} italic_τ start_POSTSUBSCRIPT Y end_POSTSUBSCRIPT ,

where 𝜷 𝜷 \bm{\beta} bold_italic_β is a vector whose components represent experimental parameters (including α 𝛼 \alpha italic_α ) that determine the gravitational signal and A 𝐴 A italic_A and B 𝐵 B italic_B are parameters encoding systematic error in the torque calibration. Multiple measurements at different locations s 𝑠 s italic_s distinguish the Yukawa torque from the Newtonian torque, inferring a best fit value for α 𝛼 \alpha italic_α . However, each torque measurement will contain uncorrelated error due to thermal torque noise with standard deviation δ ⁢ N 𝛿 𝑁 \delta N italic_δ italic_N , such that a nonzero estimate of α 𝛼 \alpha italic_α may be inferred even in the absence of a Yukawa force. The expected variance of this estimate is used to define the minimum detectable interaction strength α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT to produce a statistically significant signal (see Appendix A ).

Generally, the Newtonian τ G subscript 𝜏 G \tau_{\text{G}} italic_τ start_POSTSUBSCRIPT G end_POSTSUBSCRIPT and Yukawa τ Y subscript 𝜏 Y \tau_{\text{Y}} italic_τ start_POSTSUBSCRIPT Y end_POSTSUBSCRIPT signals depend on experimental parameters 𝜷 𝜷 \bm{\beta} bold_italic_β , such as those parametrizing the shape, size, or alignment of the source and test masses. In practice, the value of each parameter may not be known exactly and these uncertainties affect α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT (see Appendix A for more details). In Fig. 2 b we account for uncertainty in the paddle widths δ ⁢ w x = δ ⁢ w y = 10 ⁢ μ 𝛿 subscript 𝑤 𝑥 𝛿 subscript 𝑤 𝑦 10 μ \delta w_{x}=\delta w_{y}=10~{}\upmu italic_δ italic_w start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = italic_δ italic_w start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT = 10 roman_μ m and uncertainty in the smallest separation δ ⁢ s 0 = 5 ⁢ μ 𝛿 subscript 𝑠 0 5 μ \delta s_{0}=5~{}\upmu italic_δ italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 5 roman_μ m. However, the limits presented are dominated by uncertainty in the torque signal calibration, where we have assumed an overall scale factor uncertainty of δ ⁢ A = 25 % 𝛿 𝐴 percent 25 \delta A=25\% italic_δ italic_A = 25 % and no prior knowledge of the offset B 𝐵 B italic_B . Perfect knowledge of these parameters would reduce α min ⁢ ( λ = 30 ⁢ μ ⁢ m ) subscript 𝛼 min 𝜆 30 μ m \alpha_{\text{min}}\left(\lambda=30~{}\upmu\text{m}\right) italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT ( italic_λ = 30 roman_μ m ) for the 300 K, 300 K (tungsten), 4 K, and 100 mK curves in Fig. 2 b by factors of 2.1, 3.1, 3.8, and 4.5, respectively.

Figure 2 b depicts the projected minimum detectable coupling strength α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT of a torsion resonator within the 1-300 μ μ ~{}\upmu roman_μ m interaction range, including limitations from thermomechanical noise and systematic uncertainties. We find that a room temperature experiment consisting of 30, one-day-long measurements at separations in the range s ∈ [ 25 , 75 ] ⁢ μ 𝑠 25 75 μ s\in\left[25,75\right]~{}\upmu italic_s ∈ [ 25 , 75 ] roman_μ m is capable of probing new parameter space over the interval 3 ⁢ μ ⁢ m ≲ λ ≲ 36 ⁢ μ less-than-or-similar-to 3 μ m 𝜆 less-than-or-similar-to 36 μ 3~{}\upmu\text{m}\lesssim\lambda\lesssim 36~{}\upmu 3 roman_μ m ≲ italic_λ ≲ 36 roman_μ m (dashed red curve). To improve upon this result, additional measures can be taken such as cryogenic cooling to reduce thermal noise or fabricating the resonator from a denser material to amplify the Yukawa signal. These scenarios are also included in Fig. 2 , where the thermal noise variance is simply rescaled by the temperature δ ⁢ N 2 ∝ T proportional-to 𝛿 superscript 𝑁 2 𝑇 \delta N^{2}\propto T italic_δ italic_N start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∝ italic_T for cryogenic cooling (4 K and 100 mK) and an additional room temperature projection is included assuming a tungsten test mass where all other device parameters are held constant, such that the tungsten-based device would have parameters ( I 0 , f 0 , Q 0 ) = ( 3 × 10 − 14 ⁢ kg ⁢ m 2 ,  48 ⁢ Hz ,  6 × 10 7 ) subscript 𝐼 0 subscript 𝑓 0 subscript 𝑄 0 3 superscript 10 14 kg superscript m 2 48 Hz 6 superscript 10 7 (I_{0},f_{0},Q_{0})=(3\times 10^{-14}~{}\text{kg}\,\text{m}^{2},\,48~{}\text{% Hz},\,6\times 10^{7}) ( italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = ( 3 × 10 start_POSTSUPERSCRIPT - 14 end_POSTSUPERSCRIPT kg m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , 48 Hz , 6 × 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ) .

The minimum separation distance in the experiment may be limited by electrostatic coupling between the resonator and the electrostatic shield. Metallization and voltage compensation can reduce this effect  [ 24 ] , however, nonuniform potentials on the conducting surfaces will remain and present a statistical, separation-distance-dependent noise source in the form of seismic patch field coupling  [ 27 , 30 ] . Polycrystallinity, surface contamination, or variation in chemical composition produce a nonuniform surface potential, commonly referred to as “patch potentials”  [ 31 ] . Random translational or rotational motion of the oscillator along any axis causes it to sample the spatially random and anharmonic potential established by the patch fields, resulting in random forces and torques. Due to the short range of the patch fields whose length scale is assumed to be smaller than the separation distance s 𝑠 s italic_s , this effect is more pronounced at shorter distances and is expected to scale as s − 4 superscript 𝑠 4 s^{-4} italic_s start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT   [ 32 , 33 ] . The effect of this seismic patch field coupling can be reduced by decreasing the oscillator’s motion through a combination of vibration isolation and simultaneous feedback cooling  [ 34 ] of the torsional and flexural modes.

The exact limitations on our experiment posed by electrostatic effects are unknown but will likely manifest as a lower limit on the surface separation. Unsure of this limit in the proposed system, in Fig. 3 we explore the effect of varying the smallest separation distance s 0 subscript 𝑠 0 s_{0} italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT on α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT and the minimum interaction length λ min subscript 𝜆 min \lambda_{\text{min}} italic_λ start_POSTSUBSCRIPT min end_POSTSUBSCRIPT for which the experiment is sensitive to gravitational-strength Yukawa couplings ( | α | = 1 𝛼 1 \left|\alpha\right|=1 | italic_α | = 1 ). For a room temperature silicon device (red), Fig. 3 a shows that for λ = 30 ⁢ μ 𝜆 30 μ \lambda=30~{}\upmu italic_λ = 30 roman_μ m a minimum separation s 0 ≲ 45 ⁢ μ less-than-or-similar-to subscript 𝑠 0 45 μ s_{0}\lesssim 45~{}\upmu italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≲ 45 roman_μ m is needed to surpass prior constraints and s 0 ≲ 10 ⁢ μ less-than-or-similar-to subscript 𝑠 0 10 μ s_{0}\lesssim 10~{}\upmu italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≲ 10 roman_μ m is needed to achieve α min ≤ 1 subscript 𝛼 min 1 \alpha_{\text{min}}\leq 1 italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT ≤ 1 . Figure 3 b shows that new constraints on λ min subscript 𝜆 min \lambda_{\text{min}} italic_λ start_POSTSUBSCRIPT min end_POSTSUBSCRIPT would require s 0 ≲ 18 ⁢ μ less-than-or-similar-to subscript 𝑠 0 18 μ s_{0}\lesssim 18~{}\upmu italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≲ 18 roman_μ m, while a cryogenic experiment at 100 mK (purple) could potentially achieve λ min ≈ 10 ⁢ μ subscript 𝜆 min 10 μ \lambda_{\text{min}}\approx 10~{}\upmu italic_λ start_POSTSUBSCRIPT min end_POSTSUBSCRIPT ≈ 10 roman_μ m with s 0 ≈ 25 ⁢ μ subscript 𝑠 0 25 μ s_{0}\approx 25~{}\upmu italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≈ 25 roman_μ m.

As a first step toward experiment, we have fabricated and characterized a prototype microresonator with dimensions comparable to the proposed devices. For the design dimensions in Fig. 4 a, the predicted frequency, quality factor, and thermal torque noise are 80 Hz, 1.3 × 10 7 1.3 superscript 10 7 1.3\times 10^{7} 1.3 × 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT , and 10 − 19 superscript 10 19 10^{-19}~{} 10 start_POSTSUPERSCRIPT - 19 end_POSTSUPERSCRIPT N m/ Hz Hz \sqrt{\rm Hz} square-root start_ARG roman_Hz end_ARG , respectively (see Appendix B ). Calibrated readout of the resonator’s angular displacement was performed using an optical lever  [ 26 ] with a 1550 1550 1550~{} 1550 nm laser beam, and Fig. 4 b shows a ringdown measurement that confirms the design quality factor Q 0 ≈ 10 7 subscript 𝑄 0 superscript 10 7 Q_{0}\approx 10^{7} italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≈ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT . The displacement spectrum S θ subscript 𝑆 𝜃 S_{\theta} italic_S start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT reveals a lower resonance frequency f 0 = 71.5 subscript 𝑓 0 71.5 f_{0}=71.5 italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 71.5 Hz than predicted, possibly due to overestimation of the thin-film stress, for example. The apparent torque spectrum (Fig. 4 c) can be inferred as S τ = | χ | − 2 ⁢ S θ subscript 𝑆 𝜏 superscript 𝜒 2 subscript 𝑆 𝜃 S_{\tau}=\left|\chi\right|^{-2}S_{\theta} italic_S start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT = | italic_χ | start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , where the inverse mechanical susceptibility is χ − 1 ≡ ( 2 ⁢ π ) 2 ⁢ I 0 ⁢ ( f 0 2 − f 2 − i ⁢ f ⁢ f 0 / Q 0 ) superscript 𝜒 1 superscript 2 𝜋 2 subscript 𝐼 0 superscript subscript 𝑓 0 2 superscript 𝑓 2 𝑖 𝑓 subscript 𝑓 0 subscript 𝑄 0 \chi^{-1}\equiv\left(2\pi\right)^{2}I_{0}\left({f_{0}}^{2}-f^{2}-iff_{0}/Q_{0}\right) italic_χ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ≡ ( 2 italic_π ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_f start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_i italic_f italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) . As shown in Fig. 4 c, we infer S τ ≈ 10 − 17 subscript 𝑆 𝜏 superscript 10 17 \sqrt{S_{\tau}}\approx 10^{-17}~{} square-root start_ARG italic_S start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT end_ARG ≈ 10 start_POSTSUPERSCRIPT - 17 end_POSTSUPERSCRIPT N m/ Hz Hz \sqrt{\rm Hz} square-root start_ARG roman_Hz end_ARG .

The torque noise of our prototype device (Fig. 4 c) is two orders of magnitude above the predicted thermal torque noise and points to a key challenge: mitigating acceleration noise that couples to the torsion mode due to asymmetric mass loading. Specifically, the excess noise corresponds to a horizontal acceleration background of 70 n g 𝑔 g italic_g / Hz Hz \sqrt{\rm Hz} square-root start_ARG roman_Hz end_ARG near 71.5 Hz, which was confirmed with an independent accelerometer measurement (blue trace in Fig. 4 c). We note that the current device is particularly susceptible to horizontal accelerations given its relatively large ( w z = 200 ⁢ μ subscript 𝑤 𝑧 200 μ w_{z}=200~{}\upmu italic_w start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 200 roman_μ m) thickness, since S τ ∝ w z 2 ⁢ S a proportional-to subscript 𝑆 𝜏 superscript subscript 𝑤 𝑧 2 subscript 𝑆 𝑎 \sqrt{S_{\tau}}\propto{w_{z}}^{2}\sqrt{S_{a}} square-root start_ARG italic_S start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT end_ARG ∝ italic_w start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT square-root start_ARG italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG . By reducing the device thickness to 50 μ μ ~{}\upmu roman_μ m, per the proposed design, this vibration-induced torque noise would be reduced by a factor of 16. To completely remove the coupling of horizontal accelerations into the torsion mode, the device may be fabricated with the center of mass aligned with the torsion axis, which can be achieved by depositing additional material on the topside of the pad, for example.

In summary, we have proposed microscale torsion resonators  [ 26 ] as a new platform for short-range gravity experiments and modeled the expected performance given limitations from thermomechanical noise and systematic uncertainties. We find that a room temperature device has sufficient sensitivity to detect new Yukawa interactions below ∼ 36 ⁢ μ similar-to absent 36 μ {\sim 36}~{}\upmu ∼ 36 roman_μ m, assuming surface separations down to 25 μ μ ~{}\upmu roman_μ m and a 30-day-long measurement campaign. As a first step, we have fabricated a prototype device exhibiting a large mechanical quality factor of 10 7 superscript 10 7 10^{7} 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT and thermal torque of 10 − 19 superscript 10 19 10^{-19}~{} 10 start_POSTSUPERSCRIPT - 19 end_POSTSUPERSCRIPT N m/ Hz Hz \sqrt{\rm Hz} square-root start_ARG roman_Hz end_ARG . The key next steps are addressing acceleration noise and potential electrostatic interactions such that measurements can be performed near the device thermal limit. In addition to new tests of Yukawa interactions, this would enable the first measurements of the Newtonian gravitational coupling between sub-milligram objects  [ 35 ] . Looking forward, we note that microscale torsion resonators are a promising platform for exploring new physics beyond ISL deviations, such as spin-dependent interactions  [ 36 ] or quantum gravity  [ 37 ] .

We thank Patrick Egan and Charles Clark for feedback on the manuscript. This work is supported by the National Science Foundation Grants No. 2239735 and No. 2134830 and Arizona State University.

Appendix A Analysis for minimum detectable coupling strength

The proposed experiment will use the gravitational torque exerted on the torsion paddle by the source mass to infer a value α ^ ^ 𝛼 \hat{\alpha} over^ start_ARG italic_α end_ARG of the Yukawa coupling strength α 𝛼 \alpha italic_α . The inferred coupling strength α ^ ^ 𝛼 \hat{\alpha} over^ start_ARG italic_α end_ARG contains error from both stochastic noise and systematic uncertainties. The non-zero variance δ ⁢ α ^ 𝛿 ^ 𝛼 \delta\hat{\alpha} italic_δ over^ start_ARG italic_α end_ARG of this estimate affects the minimum detectable coupling strength α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT .

In this section, we provide a treatment for calculating α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT based on estimated limitations from thermal noise and systematic error due to uncertainty in experimental parameters. We also provide a brief explanation of the numerical simulations used to model the expected Newtonian and Yukawa torque signals.

A.1 Measurement and parameter uncertainty

At a given surface separation distance s i subscript 𝑠 𝑖 s_{i} italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , the source mass exerts a torque on the paddle containing both Newtonian τ G subscript 𝜏 𝐺 \tau_{G} italic_τ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT and Yukawa τ Y subscript 𝜏 𝑌 \tau_{Y} italic_τ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT contributions

subscript 𝑠 0 Δ subscript 𝑠 𝑖 s_{i}=s_{0}+\Delta s_{i} italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_Δ italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , such that the increments Δ ⁢ s i Δ subscript 𝑠 𝑖 \Delta s_{i} roman_Δ italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are treated as the independent variables and the smallest separation s 0 subscript 𝑠 0 s_{0} italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is included as a component in 𝜷 𝜷 \bm{\beta} bold_italic_β .

The actual measured data

Following the treatment in Ref. [ 15 ] , the data from an experiment with multiple measurements can be fitted to estimate the value of 𝜼 𝜼 \bm{\eta} bold_italic_η by minimizing the quantity

with respect to 𝜼 ^ bold-^ 𝜼 \bm{\hat{\eta}} overbold_^ start_ARG bold_italic_η end_ARG . Here, Latin indices refer to torque measurements and Greek indices refer to individual parameters and their independent measurements or estimates.

Our analysis in the main text accounts for uncertainty in several experimental parameters: the Yukawa interaction strength ( α 𝛼 \alpha italic_α ), the torsion resonator widths ( w x subscript 𝑤 𝑥 w_{x} italic_w start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , w y subscript 𝑤 𝑦 w_{y} italic_w start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ), and the smallest source-test mass separation distance ( s 0 subscript 𝑠 0 s_{0} italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ), such that 𝜷 ^ = ( α ^ , w ^ x , w ^ y , s ^ 0 ) bold-^ 𝜷 ^ 𝛼 subscript ^ 𝑤 𝑥 subscript ^ 𝑤 𝑦 subscript ^ 𝑠 0 \bm{\hat{\beta}}=\left(\hat{\alpha},\hat{w}_{x},\hat{w}_{y},\hat{s}_{0}\right) overbold_^ start_ARG bold_italic_β end_ARG = ( over^ start_ARG italic_α end_ARG , over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , over^ start_ARG italic_s end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and 𝜼 ^ = ( α ^ , w ^ x , w ^ y , s ^ 0 , a ^ , b ^ ) bold-^ 𝜼 ^ 𝛼 subscript ^ 𝑤 𝑥 subscript ^ 𝑤 𝑦 subscript ^ 𝑠 0 ^ 𝑎 ^ 𝑏 \bm{\hat{\eta}}=\left(\hat{\alpha},\hat{w}_{x},\hat{w}_{y},\hat{s}_{0},\hat{a}% ,\hat{b}\right) overbold_^ start_ARG bold_italic_η end_ARG = ( over^ start_ARG italic_α end_ARG , over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , over^ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , over^ start_ARG italic_s end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over^ start_ARG italic_a end_ARG , over^ start_ARG italic_b end_ARG ) . While the vector 𝜷 𝜷 \bm{\beta} bold_italic_β will more generally contain additional parameters, such as the material densities or geometric parameters of the source mass, for simplicity we assume they are known with certainty. Note that this choice of parameters means that we include uncertainty in s 0 subscript 𝑠 0 s_{0} italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , but we treat the increments Δ ⁢ s i Δ subscript 𝑠 𝑖 \Delta s_{i} roman_Δ italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT as independent variables with negligible error.

A.2 Linearized solution

and redefining the parameter vector accordingly: 𝜼 → ( β 0 , β 1 , … , A , B ) → 𝜼 subscript 𝛽 0 subscript 𝛽 1 … 𝐴 𝐵 \bm{\eta}\rightarrow\left(\beta_{0},\beta_{1},...,A,B\right) bold_italic_η → ( italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A , italic_B ) . By defining a new function

Eq. 6 transforms to

An analytical solution for the fit parameters η ^ ν subscript ^ 𝜂 𝜈 \hat{\eta}_{\nu} over^ start_ARG italic_η end_ARG start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT that minimize χ 2 superscript 𝜒 2 \chi^{2} italic_χ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT can be obtained by linearizing τ i ⁢ ( 𝜼 ^ ) subscript 𝜏 𝑖 bold-^ 𝜼 \tau_{i}(\bm{\hat{\eta}}) italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( overbold_^ start_ARG bold_italic_η end_ARG ) with respect to its parameters

Under this approximation, it can be shown that the solution for each component of 𝜼 ^ ^ 𝜼 \hat{\bm{\eta}} over^ start_ARG bold_italic_η end_ARG will have an expected variance

We define the minimum detectable coupling strength α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT in terms of the expected variance of the estimator α ^ ^ 𝛼 \hat{\alpha} over^ start_ARG italic_α end_ARG under the assumption that α = 0 𝛼 0 \alpha=0 italic_α = 0 and that α ^ ^ 𝛼 \hat{\alpha} over^ start_ARG italic_α end_ARG is not constrained by prior measurement ( δ ⁢ α = ∞ 𝛿 𝛼 \delta\alpha=\infty italic_δ italic_α = ∞ ). If the 0th parameter is α 𝛼 \alpha italic_α , i.e. η 0 = α subscript 𝜂 0 𝛼 \eta_{0}=\alpha italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_α , then for 2 σ 𝜎 \sigma italic_σ confidence

A.3 Torque simulations

We performed numerical simulations to estimate the amplitudes of the torques τ G ⁢ ( 𝜼 , Δ ⁢ s ) subscript 𝜏 G 𝜼 Δ 𝑠 \tau_{\text{G}}({\bm{\eta}},\Delta s) italic_τ start_POSTSUBSCRIPT G end_POSTSUBSCRIPT ( bold_italic_η , roman_Δ italic_s ) and τ Y ⁢ ( 𝜼 , Δ ⁢ s ) subscript 𝜏 Y 𝜼 Δ 𝑠 \tau_{\text{Y}}({\bm{\eta}},\Delta s) italic_τ start_POSTSUBSCRIPT Y end_POSTSUBSCRIPT ( bold_italic_η , roman_Δ italic_s ) exerted on the test mass by the source mass. For various interaction lengths in the range λ ∈ [ 1 ⁢ μ ⁢ m , 1 ⁢  cm ] 𝜆 1 μ m 1  cm \lambda\in\left[1~{}\upmu\text{m},1\text{ cm}\right] italic_λ ∈ [ 1 roman_μ m , 1 cm ] , the Newtonian and Yukawa potentials were calculated over a rectangular grid in the region x ∈ 𝑥 absent x\in italic_x ∈ [-550,550] μ μ ~{}\upmu roman_μ m, y ∈ 𝑦 absent y\in italic_y ∈ [-550,550] μ μ ~{}\upmu roman_μ m, z ∈ 𝑧 absent z\in italic_z ∈ [25,250] μ μ ~{}\upmu roman_μ m, using stratified Monte Carlo sampling of the source mass distribution. A numerical gradient operation was then performed to extract the vector Newtonian and Yukawa field components.

The torque on the torsion resonator was computed for a given pad geometry as a weighted (accounting for the local lever arm) numerical integration over the Newtonian and Yukawa fields. These torque calculations were repeated at various surface separations to infer the torque’s dependence on separation distance s 𝑠 s italic_s . Parameters ( w x , w y ) subscript 𝑤 𝑥 subscript 𝑤 𝑦 \left(w_{x},w_{y}\right) ( italic_w start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) were also independently swept at each separation distance in order to calculate Ψ i ⁢ ν subscript Ψ 𝑖 𝜈 \Psi_{i\nu} roman_Ψ start_POSTSUBSCRIPT italic_i italic_ν end_POSTSUBSCRIPT . In order to estimate α min subscript 𝛼 min \alpha_{\text{min}} italic_α start_POSTSUBSCRIPT min end_POSTSUBSCRIPT for any arbitrary set of chosen separation distances s i subscript 𝑠 𝑖 s_{i} italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , polynomial fits were performed to approximate the sampled Ψ i ⁢ ν subscript Ψ 𝑖 𝜈 \Psi_{i\nu} roman_Ψ start_POSTSUBSCRIPT italic_i italic_ν end_POSTSUBSCRIPT as smooth functions of separation distance.

Appendix B Models for the resonator mechanical properties

The thermal torque noise generally depends on the resonator’s frequency f 0 subscript 𝑓 0 f_{0} italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , moment of inertia I 0 subscript 𝐼 0 I_{0} italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , and quality factor Q 0 subscript 𝑄 0 Q_{0} italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT . Here we present the models used for each of these parameters. The frequency can be modeled as

where w rib subscript 𝑤 rib w_{\text{rib}} italic_w start_POSTSUBSCRIPT rib end_POSTSUBSCRIPT , L rib subscript 𝐿 rib L_{\text{rib}} italic_L start_POSTSUBSCRIPT rib end_POSTSUBSCRIPT , and h rib subscript ℎ rib h_{\text{rib}} italic_h start_POSTSUBSCRIPT rib end_POSTSUBSCRIPT are respectively the ribbon’s width, length, and thickness. Following Ref.  [ 26 ] , we assume the ribbon’s stress to be σ rib = 0.85 subscript 𝜎 rib 0.85 \sigma_{\text{rib}}=0.85 italic_σ start_POSTSUBSCRIPT rib end_POSTSUBSCRIPT = 0.85 GPa and elastic modulus to be E rib = 250 subscript 𝐸 rib 250 E_{\text{rib}}=250 italic_E start_POSTSUBSCRIPT rib end_POSTSUBSCRIPT = 250 GPa. The gravitational stiffness comes from the restoring torque exerted on the torsion pad by Earth’s gravity g 𝑔 g italic_g . When the device is oriented such that the pad hangs below the ribbon, the gravitational stiffness is

Mechanical dissipation in the resonator inherently depends on the material’s intrinsic quality factor, which for Si 3 N 4 can be modeled as  [ 39 ]

where surface loss plays a larger role in thinner films. However, due to dissipation dilution, the quality factor of the resonator Q 0 subscript 𝑄 0 Q_{0} italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is enhanced as  [ 26 ]

• Adelberger  et al. [2003] E. Adelberger, B. Heckel, and A. Nelson, Tests of the gravitational inverse-square law,  Annual Review of Nuclear and Particle Science  53 , 77 (2003) .
• Adelberger  et al. [2009] E. G. Adelberger, J. Gundlach, B. Heckel, S. Hoedl, and S. Schlamminger, Torsion balance experiments: A low-energy frontier of particle physics,  Progress in Particle and Nuclear Physics  62 , 102 (2009) .
• Kapner  et al. [2007] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle, and H. E. Swanson, Tests of the gravitational inverse-square law below the dark-energy length scale,  Phys. Rev. Lett.  98 , 021101 (2007) .
• Sundrum [2004] R. Sundrum, Fat gravitons, the cosmological constant and submillimeter tests,  Physical Review D  69 , 044014 (2004) .
• Adelberger  et al. [2007] E. G. Adelberger, B. R. Heckel, S. Hoedl, C. D. Hoyle, D. J. Kapner, and A. Upadhye, Particle-physics implications of a recent test of the gravitational inverse-square law,  Phys. Rev. Lett.  98 , 131104 (2007) .
• Upadhye  et al. [2012] A. Upadhye, W. Hu, and J. Khoury, Quantum stability of chameleon field theories,  Phys. Rev. Lett.  109 , 041301 (2012) .
• Montero  et al. [2023] M. Montero, C. Vafa, and I. Valenzuela, The dark dimension and the swampland,  Journal of High Energy Physics  2023 , 1 (2023) .
• Vafa [2024] C. Vafa, Swamplandish unification of the dark sector (2024),  arXiv:2402.00981 [hep-ph] .
• Kaplan and Wise [2000] D. B. Kaplan and M. B. Wise, Couplings of a light dilaton and violations of the equivalence principle,  Journal of High Energy Physics  2000 , 037 (2000) .
• Antoniadis  et al. [2003] I. Antoniadis, K. Benakli, A. Laugier, and T. Maillard, Brane to bulk supersymmetry breaking and radion force at micron distances,  Nuclear Physics B  662 , 40 (2003) .
• Arkani-Hamed  et al. [1999] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phenomenology, astrophysics, and cosmology of theories with submillimeter dimensions and tev scale quantum gravity,  Physical Review D  59 , 086004 (1999) .
• Dimopoulos and Geraci [2003] S. Dimopoulos and A. A. Geraci, Probing submicron forces by interferometry of bose-einstein condensed atoms,  Physical Review D  68 , 124021 (2003) .
• Hoyle  et al. [2001] C. D. Hoyle, U. Schmidt, B. R. Heckel, E. G. Adelberger, J. H. Gundlach, D. J. Kapner, and H. E. Swanson, Submillimeter test of the gravitational inverse-square law: A search for “large” extra dimensions,  Phys. Rev. Lett.  86 , 1418 (2001) .
• Hoyle  et al. [2004] C. D. Hoyle, D. Kapner, B. R. Heckel, E. Adelberger, J. Gundlach, U. Schmidt, and H. Swanson, Submillimeter tests of the gravitational inverse-square law, Physical Review D  70 , 042004 (2004).
• Lee  et al. [2020] J. Lee, E. Adelberger, T. Cook, S. Fleischer, and B. Heckel, New test of the gravitational 1/r 2 law at separations down to 52 μ μ \,\upmu roman_μ m,  Physical Review Letters  124 , 101101 (2020) .
• Geraci  et al. [2008] A. A. Geraci, S. J. Smullin, D. M. Weld, J. Chiaverini, and A. Kapitulnik, Improved constraints on non-newtonian forces at 10 microns,  Physical Review D  78 , 022002 (2008) .
• Chen  et al. [2016] Y.-J. Chen, W. K. Tham, D. Krause, D. López, E. Fischbach, and R. S. Decca, Stronger limits on hypothetical yukawa interactions in the 30–8000 nm range,  Physical Review Letters  116 , 221102 (2016) .
• Blakemore  et al. [2021] C. P. Blakemore, A. Fieguth, A. Kawasaki, N. Priel, D. Martin, A. D. Rider, Q. Wang, and G. Gratta, Search for non-newtonian interactions at micrometer scale with a levitated test mass,  Physical Review D  104 , L061101 (2021) .
• Fischbach  et al. [2001] E. Fischbach, D. Krause, V. Mostepanenko, and M. Novello, New constraints on ultrashort-ranged Yukawa interactions from atomic force microscopy,  Physical Review D  64 , 075010 (2001) .
• Mostepanenko and Novello [2001] V. Mostepanenko and M. Novello, Constraints on non-Newtonian gravity from the Casimir force measurements between two crossed cylinders,  Physical Review D  63 , 115003 (2001) .
• Tu  et al. [2007] L.-C. Tu, S.-G. Guan, J. Luo, C.-G. Shao, and L.-X. Liu, Null test of newtonian inverse-square law at submillimeter range with a dual-modulation torsion pendulum,  Physical review letters  98 , 201101 (2007) .
• Cook [2013] T. Cook,  A test of the gravitational inverse-square law at short distance ,  Ph.D. thesis , University of Washington (2013).
• Tan  et al. [2016] W.-H. Tan, S.-Q. Yang, C.-G. Shao, J. Li, A.-B. Du, B.-F. Zhan, Q.-L. Wang, P.-S. Luo, L.-C. Tu, and J. Luo, New test of the gravitational inverse-square law at the submillimeter range with dual modulation and compensation,  Physical Review Letters  116 , 131101 (2016) .
• Tan  et al. [2020] W.-H. Tan, A.-B. Du, W.-C. Dong, S.-Q. Yang, C.-G. Shao, S.-G. Guan, Q.-L. Wang, B.-F. Zhan, P.-S. Luo, L.-C. Tu, et al. , Improvement for testing the gravitational inverse-square law at the submillimeter range,  Physical Review Letters  124 , 051301 (2020) .
• Weld  et al. [2008] D. M. Weld, J. Xia, B. Cabrera, and A. Kapitulnik, New apparatus for detecting micron-scale deviations from newtonian gravity,  Physical Review D  77 , 062006 (2008) .
• Pratt  et al. [2023a] J. R. Pratt, A. R. Agrawal, C. A. Condos, C. M. Pluchar, S. Schlamminger, and D. J. Wilson, Nanoscale torsional dissipation dilution for quantum experiments and precision measurement,  Physical Review X  13 , 011018 (2023a) .
• Dong  et al. [2023] W.-C. Dong, W.-H. Tan, Z.-J. An, H. Huang, L. Zhu, Y.-J. Tan, T.-Y. Long, C.-G. Shao, and S.-Q. Yang, Coupling effect of vibrations and residual electrostatic force in short-range gravitational experiments,  Physical Review Applied  20 , 054046 (2023) .
• Ke  et al. [2023] J. Ke, W.-C. Dong, S.-H. Huang, Y.-J. Tan, W.-H. Tan, S.-Q. Yang, C.-G. Shao, and J. Luo, Electrostatic effect due to patch potentials between closely spaced surfaces,  Physical Review D  107 , 065009 (2023) .
• Yu  et al. [2012] P.-L. Yu, T. Purdy, and C. Regal, Control of material damping in high-q membrane microresonators,  Physical review letters  108 , 083603 (2012) .
• Lee [2020] J. G. Lee,  A Fourier-Bessel test of the gravitational inverse-square law ,  Ph.D. thesis , University of Washington (2020).
• Speake and Trenkel [2003] C. Speake and C. Trenkel, Forces between conducting surfaces due to spatial variations of surface potential,  Physical Review Letters  90 , 160403 (2003) .
• Behunin  et al. [2012] R. Behunin, F. Intravaia, D. Dalvit, P. M. Neto, and S. Reynaud, Modeling electrostatic patch effects in casimir force measurements,  Physical Review A  85 , 012504 (2012) .
• Turchette  et al. [2000] Q. A. Turchette, B. King, D. Leibfried, D. Meekhof, C. Myatt, M. Rowe, C. Sackett, C. Wood, W. Itano, C. Monroe, et al. , Heating of trapped ions from the quantum ground state,  Physical Review A  61 , 063418 (2000) .
• Poggio  et al. [2007] M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, Feedback cooling of a cantilever’s fundamental mode below 5 mk,  Phys. Rev. Lett.  99 , 017201 (2007) .
• Westphal  et al. [2021] T. Westphal, H. Hepach, J. Pfaff, and M. Aspelmeyer, Measurement of gravitational coupling between millimetre-sized masses,  Nature  591 , 225 (2021) .
• Terrano  et al. [2015] W. Terrano, E. Adelberger, J. Lee, and B. Heckel, Short-range, spin-dependent interactions of electrons: A probe for exotic pseudo-goldstone bosons,  Physical review letters  115 , 201801 (2015) .
• Carney  et al. [2021] D. Carney, H. Müller, and J. M. Taylor, Using an atom interferometer to infer gravitational entanglement generation,  PRX Quantum  2 , 030330 (2021) .
• Pratt  et al. [2023b] J. R. Pratt, S. Schlamminger, A. R. Agrawal, C. A. Condos, C. M. Pluchar, and D. J. Wilson, The intersection of noise, amplitude, and nonlinearity in a high-q micromechanical torsion pendulum, in  International Conference on Nonlinear Dynamics and Applications  (Springer, 2023) pp. 3–14.
• Villanueva and Schmid [2014] L. G. Villanueva and S. Schmid, Evidence of surface loss as ubiquitous limiting damping mechanism in sin micro-and nanomechanical resonators,  Physical Review Letters  113 , 227201 (2014) .