pendulum experiment materials

16.4 The Simple Pendulum

Learning objectives.

By the end of this section, you will be able to:

• Measure acceleration due to gravity.

Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 16.13 . Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period.

We begin by defining the displacement to be the arc length s s . We see from Figure 16.13 that the net force on the bob is tangent to the arc and equals − mg sin θ − mg sin θ . (The weight mg mg has components mg cos θ mg cos θ along the string and mg sin θ mg sin θ tangent to the arc.) Tension in the string exactly cancels the component mg cos θ mg cos θ parallel to the string. This leaves a net restoring force back toward the equilibrium position at θ = 0 θ = 0 .

Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about 15º 15º ), sin θ ≈ θ sin θ ≈ θ ( sin θ sin θ and θ θ differ by about 1% or less at smaller angles). Thus, for angles less than about 15º 15º , the restoring force F F is

The displacement s s is directly proportional to θ θ . When θ θ is expressed in radians, the arc length in a circle is related to its radius ( L L in this instance) by:

For small angles, then, the expression for the restoring force is:

This expression is of the form:

where the force constant is given by k = mg / L k = mg / L and the displacement is given by x = s x = s . For angles less than about 15º 15º , the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.

Using this equation, we can find the period of a pendulum for amplitudes less than about 15º 15º . For the simple pendulum:

for the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. As with simple harmonic oscillators, the period T T for a pendulum is nearly independent of amplitude, especially if θ θ is less than about 15º 15º . Even simple pendulum clocks can be finely adjusted and accurate.

Note the dependence of T T on g g . If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. Consider the following example.

Example 16.5

Measuring acceleration due to gravity: the period of a pendulum.

What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?

We are asked to find g g given the period T T and the length L L of a pendulum. We can solve T = 2π L g T = 2π L g for g g , assuming only that the angle of deflection is less than 15º 15º .

• Square T = 2π L g T = 2π L g and solve for g g : g = 4π 2 L T 2 . g = 4π 2 L T 2 . 16.30
• Substitute known values into the new equation: g = 4π 2 0 . 75000 m 1 . 7357 s 2 . g = 4π 2 0 . 75000 m 1 . 7357 s 2 . 16.31
• Calculate to find g g : g = 9 . 8281 m / s 2 . g = 9 . 8281 m / s 2 . 16.32

This method for determining g g can be very accurate. This is why length and period are given to five digits in this example. For the precision of the approximation sin θ ≈ θ sin θ ≈ θ to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about 0.5º 0.5º .

Making Career Connections

Knowing g g can be important in geological exploration; for example, a map of g g over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.

Take Home Experiment: Determining g g

Use a simple pendulum to determine the acceleration due to gravity g g in your own locale. Cut a piece of a string or dental floss so that it is about 1 m long. Attach a small object of high density to the end of the string (for example, a metal nut or a car key). Starting at an angle of less than 10º 10º , allow the pendulum to swing and measure the pendulum’s period for 10 oscillations using a stopwatch. Calculate g g . How accurate is this measurement? How might it be improved?

Check Your Understanding

An engineer builds two simple pendula. 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 10 kg . Pendulum 2 has a bob with a mass of 100 kg 100 kg . Describe how the motion of the pendula will differ if the bobs are both displaced by 12º 12º .

The movement of the pendula will not differ at all because the mass of the bob has no effect on the motion of a simple pendulum. The pendula are only affected by the period (which is related to the pendulum’s length) and by the acceleration due to gravity.

PhET Explorations

Pendulum lab.

Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It’s easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of g g on planet X. Notice the anharmonic behavior at large amplitude.

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Pendulum Experiment

The Pendulum Experiment is an experiment about gravity. Pendulums (or pendula if we are being exact!) are a fascinating scientific phenomenon.

This article is a part of the guide:

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For many years they have been used for keeping time. If you pull back a pendulum and then let it go, the time it takes to swing over and then return back to its starting position is one period.

They follow some simple mathematical rules and we are going to find out how they work.

We are going to do a series of three experiments to see what effect changing things has on a pendulum.

Please note that this experiment is probably easier with more than one person.

Facts About Pendulums

Pendulums have been around for thousands of years. The ancient Chinese used the pendulum principle to try and help predict earthquakes.

Galileo Galilei was the first European to really study pendulums and he discovered that their regularity could be used for keeping time, leading to the first clocks

In 1656, the Dutch inventor and mathematician, Huygens, was the first man to successfully build an accurate clock.

What You Will Need for the Pendulum Experiment

A long piece of string, at least 1 meter long.

One piece of metal wire to bend into a hook.

Some nuts from a toolbox - they must all be the same weight and must fit onto the hook.

A large piece of paper to put behind the pendulum or a wall that nobody minds you drawing on.

A stopwatch.

Initial Setting Up the Pendulum Experiment

To do this experiment requires a little building work but nothing too complicated.

The pencil should be firmly taped to the top of the tabled, leaving about 4cm hanging over the edge.

Next make a loop in your string to fit on the end of the pencil but do not make it too tight fitting.

At the other end of your string tie your hook and slide one of the nuts onto the hook.

Put your piece of card flat behind the pendulum and you are ready to go.

Before performing the pendulum experiment , make sure that everything swings freely without sticking.

Experiment One - Changing the Weight

In this experiment we are going to find out what effect changing the mass on the end of the string makes

Take your string back about 40 - 50 cm. You must make a mark on the wall or your piece of paper to make sure that you let it go from the same place every time.

As you let it go, start the stop-watch, and count the number of oscillations in one minute

Repeat the experiment 5 times and calculate an average

Put another weight on the hook

Release the weight from exactly the same place. Calculate the period as before.

Repeat 5 times and average the results

Try the same procedure with after adding another weight

You may be surprised by your results!

Experiment Two - Changing the Angle

Go back to just one weight on the string

You have the results from the first mark in your last experiment so you can use these results again.

Now, take the string back only about 20cm and make a mark as before

Let go and count the number of periods for one minute

Repeat 5 times and then work out an average

Try exactly the same thing but now let go from 10cm.

What difference does the angle of swing make?

Experiment Three - Changing the Length of the String

You already have your results from the first experiment and can use these again.

Take the string of the pendulum and cut off about 20cm. If you are really organized, you can use another length of string from the same roll to make a shorter one.

Take back to the same angle and let it fly.

Take another 20cm off the string, replace and try again.

What effect does changing the length of the string have on a pendulum?

As you can see from your results, changing a few things on a pendulum can have some unexpected effects.

There are still more questions about pendulums. What makes them slow down and stop? How does a pendulum in a grandfather clock keep swinging for a long time?

Maybe your next experiment could answer some of these questions.

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Simple Pendulum

Definition: what is a simple pendulum.

A pendulum is a device that is found in wall clocks. It consists of a weight (bob) suspended from a pivot by a string or a very light rod so that it can swing freely. When displaced to an initial angle and released, the pendulum will swing back and forth with a periodic motion. By applying Newton’s second law of motion for rotational systems, the equation of motion for the pendulum may be obtained.

Although the pendulum has a long history, Italian scientist Galileo Galilei was the first to study the properties of pendulums, beginning around 1602.

Terms Associated with Simple Pendulum

Length (L): Distance between the point of suspension to the center of the bob

Time Period (T): Time taken by the pendulum to finish one full oscillation

Linear Displacement (x): Distance traveled by the pendulum bob from the equilibrium position to one side.

Angular Displacement (θ) : The angle described by the pendulum with an imaginary axis at the equilibrium position is called the angular displacement.

Amplitude (x max ): Maximum distance traveled by the pendulum from the equilibrium position to one side before changing its direction. For angle, it is denoted by θ max .

Equation of Simple Pendulum

How to derive the formula for time period.

According to Newton’s second law,

The equation can be written in differential form as

If the amplitude of displacement is small, then the small-angle approximation holds, i.e., sin θ ~ θ.

This equation represents a simple harmonic motion. Thus, the motion of a simple pendulum is a simple harmonic motion with an angular frequency, \( \omega = \sqrt{\frac{g}{L}} \) , and linear frequency, \( f = \frac{1}{2\pi}\sqrt{\frac{g}{L}} \) . The time period is given by,

Performing dimension analysis on the right side of the above equation gives the unit of time.

[L/LT -2 ] 1/2 = [T]

The principle of a simple pendulum can be understood as follows. The restoring force of the pendulum from the above is, F = -mgL θ. This force is responsible for restoring the pendulum to its equilibrium position. However, due to the inertia of motion, the pendulum passes the equilibrium position and swings to the other side. This motion is periodic and can be solved using differential equation analysis.

After solving the differential equation, the angular displacement is given by

θ = θ max sin (ωt)

Sometimes, a phase φ is added to the above equation depending upon the initial conditions of the pendulum. Then, the equation can be written as

θ = θ max sin (ωt + φ)

A simple pendulum is a typical laboratory experiment in many academic curricula. Students are often asked to evaluate the value of the acceleration due to gravity, g, using the equation for the time period of a pendulum. Rearranging the time period equation,

Note that the component mg cos θ is balanced by the tension T of the string, i.e., T = mg cos θ.

Laws of Simple Pendulum

• Law of mass: The time period is independent of the mass of the bob.
• Law of length: The time period is directly proportional to the square root of the length.
• Law of Iscochronism: The time period is independent of the amplitude as long as the amplitude is small.
• Law of gravity: The time period is inversely proportional to the square root of the acceleration due to gravity at that place.

Uses and Applications of the Simple Pendulum

• Pendulum clock – A common household item. Every time the pendulum swings, the clock’s hand advances at a fixed rate, thus giving the time.
• Old seismometers – A pendulum with a stylus at its bottom was connected to a frame. During an earthquake, the frame moves and causes the stylus to form a pattern on paper.
• Pendulum gravimeter – A pendulum is used to measure the local gravity.
• Foucault’s pendulum – A device to measure the rotation of the earth.
• Metronome – A device used by musicians. It emits a click or a light for each beat of a predetermined interval.
• The Simple Pendulum – Acs.psu.edu
• Simple Pendulum – Hyperphysics.phy-astr.gsu.edu
• The Simple Pendulum (Simple Harmonic Motion) – Deanza.edu
• The Simple Pendulum – Iu.pressbooks.pub
• Applications of Differential Equations – Calculuslab.deltacollege.edu
• Real-world applications of Pendulums – Sites.google.com
• The Use of Pendulums in the Real World – Sciencing.com
• Oscillation of a Simple Pendulum – Acs.psu.edu
• Simple pendulum – Amrita.olabs.edu.in

Article was last reviewed on Saturday, September 30, 2023

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The Simple Pendulum Experiment

Cotton thread, small pendulum bob, protractor, split cork, metre ruler, stopwatch, piece of card with a drawn line, retort stand/clamp/boss.

1. Set up a small simple pendulum, as shown in the diagram.

5. Estimate the uncertainty in your result.

Timing Swings:

At the end of a swing the bob is stationary for a brief instant before swinging back again. It is therefore difficult to identify the precise instant of the end of a swing since the bob is far out for a long time. At the centre of the swing the bob is moving rapidly, and the instant that it passes the central point is more precisely defined. Technically timing the swings from the central position is better. It is common to use a mark to assist in identifying the instant when the bob passes the centre position (e.g. a line on a piece of card). To find the time of one swing you should time a number of swings so that the total time taken is at least 30 seconds and you should repeat this measurement at least once more.

The Pendulum Lab

The purpose of this lab is to investigate and verify (or refute) the theoretical relationships among period, length, mass, and amplitude of a pendulum by graphical analysis.

Theory and Definitions

A simple pendulum consists of a small object (the “bob”) suspended by a lightweight cord.   The mass of the pendulum is actually only the mass of the bob; the mass of the string is not included.   The period of a pendulum is the amount of time for the bob to complete exactly one cycle or oscillation back and forth.   The length of the pendulum extends from the attached end of the string to the center of mass of the bob.   The amplitude of the pendulum is the angle formed by a vertical line and the cord when the bob reaches its maximum outward displacement.  

As long as it does not swing too far from center, a simple pendulum exhibits a special type of behavior known in physics as simple harmonic motion (SHM).   See pp. 309 – 320 of your book for more details about SHM.   According to theory, the period of a simple pendulum will not depend on the mass of the bob or the amplitude (so long as it is less than about 10 ° ).   Instead, the period will depend only upon the length as indicated in the following equation:

where:   T = period, l = length, and g = a gravitational constant

Procedural Notes

Suspend the bob from a ring stand using string.   Rigidly attach a meter stick to the ring stand such that the zero on the scale is aligned with the point where the string is attached.   When measuring the length read the value at the point where the bob just touches the meter stick (which should be at the center of the bob).   Attach a protractor to the meter stick such that the vertex is aligned with the point where the string is attached.   When determining the period of the pendulum measure the total time required for a certain number of cycles and then divide by that number.   In most cases it is convenient and sufficient to measure ten cycles.   The greater the number of cycles that you measure the greater should be the accuracy and precision of the calculated period.   Let the bob start swinging before you do the timing of the ten cycles; in other words do not try to release the bob and start the stopwatch simultaneously.  

You will collect data as a group and each person in your group will use the same data to prepare the lab report.   However, everything in your report except the data should be your own unique work.   Lab reports are not a group project.   Each group is to complete Part A and then choose either Part B or Part C (but not both).

A.   Period vs. Length

Measure and record the mass of the bob.   Record the amplitude as less than or equal to 10 ° .   Make a data table with 5 columns:   length, time, # of cycles, period, and k (the constant in the best fit equation – to be completed later during your analysis).   In this part of the experiment you will vary the length of the pendulum and measure the resulting period (without changing the mass or amplitude).   It is important that the amplitude of the bob’s oscillation remain less than or equal to 10 ° - the smaller the better.   Start with a length of 100.0 cm and decrease in steps of 10 cm until you reach a length of 10.0 cm.   Then collect one more datum with a length of 5.0 cm.   This will give you a total of eleven rows of data in your table.

Choose only one of the following:

B. Period vs. Mass

Measure and record the length of the bob and keep it at a constant value (try around 25 cm).   Record the amplitude as less than or equal to 10 ° .   Make a data table with 4 columns:   mass, time, # of cycles, and period.   In this part of the experiment you will vary the mass of the bob and measure the resulting period (without changing the length or amplitude).   It is important that the amplitude of the bob’s oscillation remain less than or equal to 10 ° - the smaller the better.   Start with the mass of your original bob.   Then attach bobs of various masses and measure the resulting period of each.

C. Period vs. Amplitude

Measure and record the length and mass of the bob and keep both constant.   Make a data table with 4 columns:   amplitude, time, # of cycles, and period.   In this part of the experiment you will vary the size of the bob’s oscillation and measure the resulting period (without changing the length or the mass).   Start at 10 ° and increase to 90 ° in increments of 10 ° .   This will give you a total of nine rows of data in your table.   Note:   As you attempt to measure greater and greater amplitudes you may find it difficult to complete 10 cycles as you time the period.   In such a case you may have to use a lesser number of cycles in your timing – use your best judgment and some common sense.

Interpretations of the Data (Analyses)

1.       For each part of the lab performed (part A and part B or C) make an appropriate and well-labeled graph.   On each graph draw the best-fit line or curve and determine the equation that best represents the relation between the variables.   Show all work on the graph itself.   Remember you must include units in all work!

2.       Using the data collected in part A, prepare a table of period and square root of the length.   Make a graph of period vs. square root of length.   Consider it to be linear, draw the best line, and determine the equation.   As always show all work on the graph. Note:   For this first lab report I want you to do the graphical analyses “by hand” – in other words do not use a graphing calculator to find the best fitting equations.   I expect to see work showing how you determined each equation.

Questions   (Write responses on separate paper!   Questions 1 and 4 require complete sentences.)

1.       How do your graphs and results support or refute the theory of a simple pendulum?   Your purpose here is to discuss , in writing, what evidence there is that indicates (or does not indicate) that the theory is correct.   Do not do any calculations for this question.   Instead, refer to the shapes, features, and types of relations and equations shown on the graph(s).   Be sure to discuss each part of the lab.   (3)

2.       Your work with the pendulum is an indirect way to measure g , an important constant related to the strength of gravity.   The accepted value for g is 9.80 m/s 2 .   (a) Calculate the value of g based on your results using the equation:   g = 4π 2 / k 2 , where k = the constant derived from your data in part A.   (b) Repeat using the slope form the period vs. square root of length graph (which should be about the same value as k ).   (c) Find the relative error for each of these two values of g .   (2 ea)

4.       Write an intelligent, grammatically correct, and concise discussion of error in this lab.   Any complete discussion of error will include:   indication or evidence of error(s) and speculation or explanation of the most likely sources of the same.   Remember to consider both types of error – random and systematic.   Your goal here is to satisfactorily explain how and why your results are not perfect.   (3)

A complete report (50 pts.) shall consist of the following– neatly labeled and in this order :

Part A data table   (4)

Part A graph – Period vs. Length (with best fit and equation)   (8)  

One of B or C:   (12)

  Part B data table

  Part B graph – Period vs. Mass (with best fit and equation)

  Part C data table

  Part C graph – Period vs. Amplitude (with best fit and equation)

Data table – Period vs. Square Root of Length   (4)

Graph – Period vs. Square Root of Length (with best fit and equation)   (8)

Answers to Questions 1 – 4   (point values given above)

What apparatus are needed for simple pendulum experiment?

The apparatus for this experiment consists of a support stand with a string clamp, a small spherical ball with a 125 cm length of light string, a meter stick, a vernier caliper , and a timer. The apparatus is shown in Figure 2.

Table of Contents

What is the purpose of the pendulum lab experiment?

The goal of this experiment was to determine the effect of mass and length on the period of oscillation of a simple pendulum . Using a photogate to measure the period, we varied the pendulum mass for a fixed length, and varied the pendulum length for a fixed mass.

How do you do a simple pendulum experiment?

• Take your string back about 40 – 50 cm.
• As you let it go, start the stop-watch, and count the number of oscillations in one minute.
• Repeat the experiment 5 times and calculate an average.
• Put another weight on the hook.
• Release the weight from exactly the same place.

What affects the period of a pendulum lab?

This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass.

What is a pendulum apparatus?

a device to measure the acceleration of gravity by a relative method.

Which devices are used to measure length of the pendulum?

The length of a simple pendulum is determined by measuring the length of thread l by meter scale and radius of bob r by Vernier callipers.

What is the conclusion of the pendulum experiment?

The surprising conclusion – the pendulum traverses a longer distance in a shorter time, than in a shorter distance, and its period is shorter. There are a number of reasons why Galileo thought that the period remains constant.

What is the conclusion of compound pendulum experiment?

Conclusion:After carrying out the compound pendulum experiment, it can be concluded that as thedistance towards the centre of gravitation on the metal bar decreases, the time intervalhowever increases making the relationship inversely proportional .

What is the hypothesis of the pendulum experiment?

The hypothesis in this case would be, “changing the amount of weight at the end of the pendulum will change the period of the pendulum.” The experiment design would involve measuring the period with one amount of weight, changing the weight without changing any other variables, then measuring the period with the new …

How do you make a pendulum for a school project?

• Tie a weight onto the end of the piece of string.
• Make a loop at the other end of the string.
• Screw the hook into the piece of wood and balance the wood between the backs of two chairs.
• Put the loop of string over the hook.
• Start the weight swinging and time how long it takes to make 20 swings.

What are the variables in a pendulum experiment?

The independent variable was the length of the pendulum and the dependant variable was the period. The controlled variables were the mass of the pendulum and the point of release.

How do you draw a pendulum diagram?

1 To make a simple Yes At the top center, draw an oblong box and another at the bottom center. In the small left circle, write “Yes.” In the right small circle, write, “No.” In the top oblong, write, “Not at this time.” In the bottom oblong, write, “Maybe.” Use a “Yes/No” pendulum chart for simple questions.

Why does increasing length increase period of pendulum?

A pendulum swinging through a large angle is being pulled down by gravity for a longer part of its swing than a pendulum swinging through a small angle, so it speeds up more, covering the larger distance of its big swing in the same amount of time as the pendulum swinging through a small angle covers its shorter …

What are the two main things that affect a pendulum’s period?

The mass and angle are the only factors that affect the period of a pendulum.

How can the accuracy of a pendulum experiment be improved?

Improve the accuracy of a measurement of periodic time by: making timings by sighting the bob past a fixed reference point (called a fiducial point ) sighting the bob as it moves fastest past a reference point. The pendulum swings fastest at its lowest point and slowest at the top of each swing.

What are the 3 parts of a pendulum?

A pendulum consists of only a few components including a length of string or wire, a bob or some type of weight and a fixed point. They can be used to prove that the planet rotates on an axis.

What is the principle of the pendulum?

By Archimedes’ principle the effective weight of the bob is reduced by the buoyancy of the air it displaces, while the mass (inertia) remains the same, reducing the pendulum’s acceleration during its swing and increasing the period.

How can a pendulum be used to measure gravity?

The pendulum method is a method of calculating the acceleration of gravity using a solid ball that connected to a rope attached to a stative pole. The pendulum is swung at a small angle resulted a simple harmonic motion .

What are the 2 devices used for measuring length?

Devices that are commonly used for measuring length are metre scale and measuring tape.

How do we measure time using pendulum?

What is the name of the equipment used to measure length?

Length, or the distance between two lengths: Tools used to measure length include a ruler, a Vernier caliper, and a micrometer screw gauge. Vernier calipers and micrometer screw gauges are more precise and can be used to measure the diameter of objects like pipe and wire.

What is the error in the simple pendulum experiment?

In a simple pendulum experiment , the maximum percentage error in the measurement of length is 2% and that in the observation of the time-period is 3%.

What is the objective for simple pendulum?

OBJECTIVES : To investigate the functional dependence of the period of a pendulum (τ) on the length of a pendulum (L), the mass of the bob (m) and the starting angle (θo).

Why is the pendulum scientifically very important?

The pendulum is important to measure the effects of gravity. Galileo observed that the reason a pendulum moves back towards the resting position is due to the gravitational pull of the earth. The spinning of the earth can be demonstrated using a pendulum.

Which value is calculated in compound pendulum experiment?

body about an axis parallel with axis of oscillation and passing through the center of gravity G. By determining L, and graphically for a particular value of T, the acceleration due to gravity g at that place and the radius of gyration K of the compound pendulum can be determined.

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Simple Pendulum: Theory, Experiment, Types & Derivation

Simple Pendulum: A simple pendulum device is represented as the point mass attached to a light inextensible string and suspended from a fixed support. A simple pendulum shows periodic motion, and it occurs in the vertical plane and is mainly driven by the gravitational force.

Ever wondered why an oscillating pendulum doesn’t slow down? Or what will happen to the time period of the simple pendulum when the displacement of the bob is increased? Will it increase as the distance required to cover to complete the oscillation increases, or will it decrease as the speed at the mean position increases, or will the speed compensate for the increased distance leaving the time period unchanged? What is the difference between a physical pendulum and a simple pendulum? There are a lot of questions about the motion of a simple pendulum. Let’s read further to find out the answers.

What is Called a Simple Pendulum?

A simple pendulum is a mechanical system of mass attached to a long massless inextensible string that performs oscillatory motion. Pendulums were used to keep a track of time in ancient days. The pendulum is also used for identifying the beats.

SHM or Simple Harmonic Motion

SHM or simple harmonic motion is the type of periodic motion in which the magnitude of restoring force on the body performing SHM is directly proportional to the displacement from the mean position but the direction of force is opposite to the direction of displacement. For SHM, \(F = – K{x^n}\) The value of ‘\(n\)’ is \(1\).

Thus the acceleration of the particle is given by, \(a = \frac{F}{m}\) \(a = \frac{{ – Kx}}{m}\) Where, \(m\) is the mass of the particle. Let \({\omega ^2} = \frac{K}{m}\) As, \(\frac{K}{m}\) is a positive constant. \( \Rightarrow \,\,a = – {\omega ^2}x\) \(\omega \) is known as angular frequency of the SHM. The time period of the Simple harmonic motion is given by, \(T = \frac{{2\pi }}{\omega }\)

Following are examples of example of the simple pendulums:

Oscillating Simple Pendulum: Calculation of Time Period

It is interesting to note that the oscillation of a simple pendulum can only be considered to be a simple harmonic motion when the oscillation is small or the amplitude of oscillation is very small as compared to two lengths of the string then by using small-angle approximation the motion of a simple pendulum is considered a simple harmonic motion. When the bob is displaced by some angle then the pendulum starts the periodic motion and for small value of angle of displacement the periodic motion is simple harmonic motion with the angular displacement of the bob.

Practice Exam Questions

\(F = mg\,{\rm{sin}}\left( \theta \right)\) \(a = g\,{\rm{sin}}\left( \theta \right)\) Here \(g\) is acceleration due to gravity. For small oscillation, \(\theta \) will be small, \({\rm{sin}}\left( \theta \right) = \theta = \frac{x}{l}\) Here \(x\) is the very small linear displacement of the bob corresponding to the displaced angle. \( \Rightarrow \,\,a = g\theta \) \( \Rightarrow \,\,a = g\frac{x}{l}\) Thus the angular frequency is given by, \( \Rightarrow \,\,{\omega ^2} = \frac{g}{l}\) The time period of the pendulum is given by, \(T = \frac{{2\pi }}{\omega }\) \( \Rightarrow \,\,T = 2\pi \sqrt {\frac{l}{g}} \) Thus from the expression for a time period of a simple pendulum, we can infer that the time period does not depend on the mass of the Bob at nor varies with the change in the small amplitude of the oscillation it only depends on the length of the string and acceleration due to this property it was widely used to keep a track of fixed interval of time does it helped the musicians to be on beats

Motion of Simple Pendulum: Effect of Gravity

As the time period of simple pendulum is given by, \(T = 2\pi \sqrt {\frac{l}{g}} \) The time period of a simple pendulum is inversely proportional to the square root of acceleration due to gravity at that point. \(T \propto \frac{1}{{\sqrt g }}\) Therefore, if the acceleration due to gravity increases the time period of the simple pendulum will decrease whereas if the acceleration due to gravity decreases the time. All the simple pendulum increases.

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Calculation of Gravity

Acceleration due to gravity can be measured with the help of a simple experiment, The period \(T\) for a simple pendulum does not depend on the mass or the initial angular displacement but depends only on the length \(L\) of the string and the value of the acceleration due to gravity. Acceleration due to gravity is given by, \(g = \frac{{4{\pi ^2}l}}{{{T^2}}}\) One cam measure the length of the string and observe the time period and the using this formula we can find the acceleration due to gravity

Physical Pendulum

For a simple pendulum, we consider the mass of the string to be negligible as compared to the Bob but for a physical pendulum, the mass of the string need not be negligible in fact any rigid body can act as a physical pendulum.

By writing the torque equation for the rigid body about the fixed point, we get the angular acceleration of the rigid body is directly proportional to the angular displacement by using small-angle approximation. External torque on the system is zero, thus, \({\tau _{{\rm{ext}}}} = 0\)

Writing torque equation about the hinged point we get, \({\tau _0} = mgl{\rm{sin}}\left( \theta  \right) = {I_{\rm{O}}}\alpha\) Solving for \(\alpha ,\) \( \Rightarrow \,\,\,\alpha = \frac{{mgl}}{{{I_{\rm{O}}}}}{\rm{sin}}\left( \theta \right)\) Using small angle approximation, \({\rm{sin}}\left( \theta \right) = \theta \) \( \Rightarrow \,\,\,\alpha = – \frac{{mgl}}{{{I_{\rm{O}}}}}\left( \theta \right)\) Thus the angular frequency is given by, \( \Rightarrow \,\,\,{\omega ^2} = \frac{{mgl}}{{{I_{\rm{O}}}}}\) Time period of a physical pendulum is given by, \(T = 2\pi \sqrt {\frac{{{I_0}}}{{mg{l_{{\rm{cm}}}}}}} \) Where, \({I_0}\) is the moment of inertia about the fixed point trough which the axis passes. \({l_{{\rm{cm}}}}\) is the distance of the centre of mass from the axis point.

Simple Pendulum Application

Simple pendulums are used in clocks as the pendulum has a fixed time period they can be used to keep a track of time. Following are example of a simple pendulum:

Pendulums can be used as metronome.

Pendulums are used to calculate acceleration due to gravity.

Sample Problems on Real Simple Pendulum

1. A simple pendulum is suspended and the bob is subjected to a constant force in the horizontal direction. Find the time period for small oscillation.

Let the magnitude of the force be, \(F.\) Let the angle at equilibrium be, \({\theta _0}\) Let the axes be along the string and perpendicular to the string,

Balancing the forces at equilibrium, \(mg{\rm{sin}}\left( {{\theta _0}} \right) = {F_0}{\rm{cos}}\left( {{\theta _0}} \right)\) \({\rm{tan}}\left( {{\theta _0}} \right) = \frac{{{F_0}}}{{mg}}\) When the pendulum is displaced by some small angle, then,

\(F = {F_0}{\rm{cos}}\left( {\theta + {\theta _0}} \right) – mg{\rm{sin}}\left( {\theta + {\theta _0}} \right)\) \( \Rightarrow F = {F_0}\left[ {{\rm{cos}}\left( {{\theta _0}} \right){\rm{cos}}\left( \theta \right) – {\rm{sin}}\left( {{\theta _0}} \right){\rm{sin}}\left( \theta \right)} \right] – mg\left[ {{\rm{sin}}\left( {{\theta _0}} \right){\rm{cos}}\left( \theta \right) + {\rm{sin}}\left( \theta \right){\rm{cos}}\left( {{\theta _0}} \right)} \right]\) For small oscillation, \({\rm{sin}}\left( \theta \right) = \theta \) \({\rm{cos}}\left( \theta \right) = 1\) \( \Rightarrow \,\,ma = {F_0}{\rm{cos}}\left( {{\theta _0}} \right) – {F_0}{\rm{sin}}\left( {{\theta _0}} \right)\theta – mg{\rm{sin}}\left( {{\theta _0}} \right) – mg{\rm{cos}}\left( {{\theta _0}} \right)\theta \) Using, \(mg{\rm{sin}}\left( {{\theta _0}} \right) = {F_0}\rm{cos}\left( {{\theta _0}} \right)\) We get, \(a = – \frac{{\left[ {{F_0}{\rm{sin}}\left( {{\theta _0}} \right) + mg{\rm{cos}}\left( {{\theta _0}} \right)} \right]}}{m}\theta \) \( \Rightarrow \,\,\,a = – \frac{{\left[ {{F_0}{\rm{sin}}\left( {{\theta _0}} \right) + mg{\rm{cos}}\left( {{\theta _0}} \right)} \right]}}{{ml}}x\) Therefore, the angular velocity is, \({\omega ^2} = \frac{{\left[ {{F_0}{\rm{sin}}\left( {{\theta _0}} \right) + mg{\rm{cos}}\left( {{\theta _0}} \right)} \right]}}{{ml}}\) Putting in the values of \({\rm{sin}}\left( {{\theta _0}} \right)\) and \({\rm{cos}}\left( {{\theta _0}} \right)\) \(\Rightarrow \,\,\,{\omega ^2} = \frac{{\left[ {\frac{{{F_0} \times {F_0}}}{{\sqrt {{{\left( {mg} \right)}^2} + {{\left( {{F_0}} \right)}^2}} }} + \frac{{mg \times mg}}{{\sqrt {{{\left( {mg} \right)}^2} + {{\left( {{F_0}} \right)}^2}} }}} \right]}}{{ml}}\) Thus the time period will be, \(T = \frac{{2\pi }}{\omega }\) \(T = 2\pi \sqrt {\frac{l}{{\sqrt {{g^2} + {{\left( {\frac{{{F_0}}}{m}} \right)}^2}} }}} \)

2. A pendulum is hanging from the roof of a bus moving with a acceleration ‘a’. Find the time period of the pendulum.

Given, The bus is moving with the acceleration ‘\(a\)’. If we apply the concept of inertial and non-inertial frame then, a pseudo force will be applied on the bob,

Let the mass of the bob be, ‘\(m\)’. Therefore, the magnitude of the pseudo force will be, \({F_p} = ma\) The direction of the pseudo force will be in the opposite direction of the acceleration of the bus, Thus if we take the resultant acceleration experienced by the simple pendulum that is the sum of gravitation acceleration \({g_{{\rm{eff}}}} = \sqrt {{g^2} + {a^2}} \) Thus, the time period of the simple pendulum is given by, \(T = 2\pi \sqrt {\frac{l}{{{g_{{\rm{eff}}}}}}} \) Therefore, the time period is, \(T = 2\pi \sqrt {\frac{l}{{\sqrt {{g^2} + {a^2}} }}} \)

A simple pendulum is a mechanical system which consists of a light inextensible string and a small bob of some mass which is made to oscillate about its mean position from left extreme to right extreme. If the displacement of the bob is small as compared to the length of the string or the angle displaced is small then the motion can be considered to be simple harmonic motion. The total energy remains constant throughout the oscillation. The kinetic energy is maximum at the mean position whereas the potential energy is maximum at the extreme positions. The physical pendulum is a mechanical system in which a rigid body is hinged and suspended from a point. For the physical pendulum, we write the torque equation instead of force as it performs angular SHM. The Time period \(T\) for a simple pendulum does not depend on the mass or the initial angular displacement but depends only on the length \(L\) of the string and the value of the acceleration due to gravity. If the effective gravitational acceleration is changed the time period of the oscillation also changes.

FAQs on Simple Pendulum

Q What is the difference between a simple pendulum and a physical pendulum? Ans: Simple pendulum is a mechanical arrangement in which bob is suspended from a point with the help of a massless, inextensible string and performs linear simple harmonic motion for small displacement whereas a physical pendulum is a rigid body hinged from a point and is to oscillate and is performs angular simple harmonic motion for small angular displacement.

Q If a simple pendulum is moving with the acceleration ‘\(g\)’ downwards , what will be the time period of the simple pendulum hanging from its roof? Ans: The effective gravity experienced by the pendulum in this particular case will be zero thus the bob will not perform a simple harmonic motion thus the time period will not be defined as it will not have a periodic motion.

Q Is energy conserved during the oscillation of a simple pendulum? Ans: Yes, in the oscillation of a simple pendulum the total energy remains conserved while the potential and the kinetic energy keep oscillating between maxima and minima with a time period of the half to that of the oscillation of the simple pendulum.

Q What will be the time period of a simple pendulum in outer space? Ans: In outer space, there will be no gravity and thus there will be no restoring force when the pendulum will be displaced thus it will not oscillate and the will be no SHM. Thus, the tie period will not be defined.

Q What type of string should be used in a simple pendulum? Ans: The string in the simple pendulum should be inextensible that is the length of the string should not change with varying force and the mass of the string should be negligible.

We hope this detailed article on Simple Pendulum helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.

Embibe wishes all the candidates the best of luck for the upcoming examination.

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Science project, pendulum waves.

Sometimes, physics can be used to create beautiful art. Kinetic art is art that relies on motion to achieve a specific effect. Often that motion is just an application of simple laws of physics. Waves and harmonic motion (some examples include pendulums and springs) are often great sources of inspiration for creating mesmerizing displays.

In this project, you will use the laws of simple pendulum motion to create a “pendulum wave apparatus”: a device where many pendulums of different lengths (and therefore different periods ) start swinging at the same time. As they move in and out of sync, the pendulums create a sequence of cycling visual wave patterns.

Build a pendulum wave apparatus.

• 2 meter sticks (or a meter stick and a ruler or tape measure)
• 3.5 meters of string (roughly)
• Nine weights that can be easily attached to a piece of string (e.g. nuts, washers, masses with hooks)
• Two stacks of books, each at least one meter high
• Cut nine pieces of strings with the following lengths: 44 cm, 41 cm, 39 cm, 37 cm, 35 cm, 33 cm, 31 cm, 30 cm, 29 cm.
• Starting at about the 10 cm mark on the meter stick, tie a piece of string at each point located every 9 cm along the ruler (roughly). Use a small piece of tape to help secure each piece of string to the stick.
• Use the two book stacks to support both ends of the meter stick. The meter stick should make a “bridge” connecting the books.
• Attach one weight to the free end of each string. Thread the string through the hole or hook on each weight and, using your ruler or tape measure, adjust the lengths to exactly match the following table, starting with the longest string. Once you get the lengths right, tie the string to the weight.

• Once you have all nine pendulums constructed, use the ruler (or a large book or even your arm) to scoop all the pendulums towards you by a few inches.
• Let all the pendulums go at once so that they swing perpendicular to the meter stick from which they are hanging.
• Enjoy the show! It’s best to watch the pendulums either from above (looking down) or from one end of the meter stick (viewed along its length). How many different patterns do you see? Does anything repeat? If any patterns do repeat, how long does it take for them to do so?

The pendulum ensemble will cycle through a number of patterns over a 30 second interval. You’ll see everything in sync, pendulum waves of varying lengths rippling down the line, alternating swings (half the pendulums going one direction, half going the other), and even seemingly chaotic motion.

The lengths of the pendulums are designed so that all of them complete a different whole number of swings every 30 seconds. The first (longest) pendulum swings 25 times in 30 seconds, the next one 26 times, the next one 27, and so on; the final (shortest) pendulum completes 33 swings in the same interval. This means that every 30 seconds, all the pendulums will swing to one side together.

Everything that happens in the middle of this interval is a stunning display of many pendulums, each with a slightly shorter period than the previous one, moving in and out of phase with one another. As the shorter pendulums start getting ahead of the longer ones, they slightly “lead” the ones next to them and create a wave effect along the meter stick. At 15 seconds—halfway through the 30-second cycle—every other pendulum (starting with the second longest) will have completed a whole number of cycles, while the remaining pendulums are all synced together at a “half cycle”. When this happens, half the pendulums are all grouped together on one side, with the remaining pendulums grouped together on the other side.

Going Further

You can modify the apparatus to add more pendulums or create a longer cycle than 30 seconds. The trick is to first decide how long you want the overall cycle to be (we used 30 seconds in this design, but it can be as long as you like). Then, decide how many times you want the longest pendulum to swing back and forth in that interval. With that in mind, the period of each successive pendulum has to be set so that it swings one more time than the previous one in the same interval.

Remember that the period of an ideal pendulum depends only on its length:

• where g = 9.8 m/s is the acceleration due to gravity,
• L is the pendulum length (in meters), and
• T is the period in seconds.

The number of times ( N ) a pendulum will swing in the overall cycle is represented by

• where T max is the overall cycle length in seconds.

Alternatively, you can use the following shortcut to design your own pendulum wave apparatus:

…which tells you the length, L(n) , of the n -th pendulum, where k is the number of cycles the whole apparatus goes through before repeating the pattern.

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Physical Pendulum Experiment • EX-5518A

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The complete solution for determining the period of a physical pendulum. This complete solution is designed for use with PASCO Capstone Software.

• 1x Large Rod Base
• 1x Stainless Steel Rod, 45 cm
• 1x Physical Pendulum Set
• 1x PASPORT Rotary Motion Sensor

See the Buying Guide for this item's required, recommended, and additional accessories.

Product Summary

In this experiment, the period of a physical pendulum, a narrow bar, is determined as a function of the distance of the pivot from the center of mass. A computer model of the system is developed, which allows the student to vary the physical parameters of the model (gravity, length, c.m. position) to match the data. This makes it possible to obtain values for the physical parameters without direct measurement. A second experiment verifies the parallel axis theorem. It also uses superposition to find the rotational inertia of a disk with an off-axis circular hole.

• Parallel Axis Theorem
• Period of a physical pendulum
• Computer modeling of a system
• Rotational inertia

What's Included

• 1x Large Rod Base (ME-8735)
• 1x Stainless Steel Rod, 45 cm (ME-8736)
• 1x Physical Pendulum Set (ME-9833)
• 1x PASPORT Rotary Motion Sensor (PS-2120A)

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This product requires PASCO software for data collection and analysis. We recommend the following option(s). For more information on which is right for your classroom, see our Software Comparison: SPARKvue vs. Capstone »

• PASCO Capstone Software

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This product requires a PASCO Interface to connect to your computer or device. We recommend the following option(s). For a breakdown of features, capabilities, and additional options, see our Interface Comparison Guide »

• 550 Universal Interface
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Experiments

Experiment Library

Perform the following experiments and more with the Physical Pendulum Experiment. Visit PASCO's Experiment Library to view more activities.

Physical Pendulum Period and Inertia

This experiment has two parts: 1. Period of a Thin Rod explores the dependence of the period of a physical pendulum (a uniform bar) on the distance between the pivot point and the center of mass of the physical pendulum. 2. In...

Physical Pendulum - Period and Inertia

Period - This experiment explores the dependence of the period of a physical pendulum (a uniform bar) on the distance between the pivot point and the center of mass of the physical pendulum. Rotational Inertia - The period of...

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EXPERIMENTAL STUDY OF DAMPED OSCILLATIONS OF A TORSIONAL PENDULUM

• Zharilkassin Iskakov Almaty University of Power Energy and Communications, Institute of Mechanics and Machine Science

Yevgrafova, N. N., & Kagan, V. L. (1970). Rukovodstvo k laboratornym rabotam po fizike, uchebnoye posobiye dlya radiotekhnicheskikh i elektropriborostroitel'nykh spetsial'nostey vuzov [Manual to laboratory works in physics, textbook for radio and electrical construction specialties of high schools]. Moscow, Russia: Higher school.

Korzun, I. N. (1991). Laboratornyye raboty po mekhanike i molekulyarnoy fizike [Laboratory work on mechanics and molecular physics]. Almaty, Kazakhstan: Kazakh National University.

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Foucault Pendulum at the South Pole: Proposal For an Experiment to Detect the Earth's General Relativistic Gravitomagnetic Field

Vladimir b. braginsky, aleksander g. polnarev, and kip s. thorne, phys. rev. lett. 53 , 863 – published 27 august 1984.

• Citing Articles (77)

An experiment is proposed for measuring the earth's gravitomagnetic field by monitoring its effect on the plane of swing of a Foucault pendulum at the south pole ("dragging of inertial frames by earth's rotation"). With great effort a 10% experiment in a measurement time of several months might be achieved.

• Received 12 March 1984

DOI: https://doi.org/10.1103/PhysRevLett.53.863

©1984 American Physical Society

Authors & Affiliations

• Physics Faculty, Moscow State University, Moscow, Union of Soviet Socialist Republics
• Space Research Institute, Academy of Sciences of the USSR, Moscow, Union of Soviet Socialist Republics
• California Institute of Technology, Pasadena, California 91125

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Vol. 53, Iss. 9 — 27 August 1984

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