simple pendulum experiment class 12

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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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Physics Article
To Find Effective Length Of Seconds Pendulum Using Graph

Using a simple pendulum, plot L-T and L-T 2 graphs and use it to find the effective length of a second’s pendulum using appropriate graph

Using a simple pendulum, plot L-T and L-T 2 graphs and use it to find the effective length of the second’s pendulum.

Apparatus and Material Required

Clamp stand
Heavy metallic spherical bob with a hook
Long and strong cotton thread
Meter scale
Graph Paper
Pencil Eraser

The simple pendulum exhibits Simple Harmonic Motion (SHM) as the acceleration of the pendulum bob is directly proportional to the displacement from the mean position and is always directed towards it. The time period (T) of a simple pendulum for oscillations of small amplitude is given by the relation

Where L is the length of the pendulum and g is the acceleration of gravity

Read More: Simple Pendulum

Place the clamp stand on the table. Tie the hook attached to the pendulum bob, to one end of the string of about 150 cm in length and the other end of the string through two half-pieces of a split cork.
Clamp the split cork firmly to the clamp stand such that the line of separation between the two pieces of the split cork is at right angles to the line OA along which the pendulum oscillates as given in the figure. Mark the edge of the table a vertical line parallel to and just behind the vertical thread OA, the position of the bob at rest. Take care that the bob hangs vertically (about 2 cm above the floor) beyond the edge of the table so that it is free to oscillate.
Measure the effective length of the simple pendulum as shown in the figure.

measure the effective length of the simple pendulum

Displace the bob not more than 15 degrees from the vertical position OA and then gently release it. If you notice the stand to be shaky, put a heavy object on its base. Make sure that the bob oscillates in a vertical plane about its rest and does not (i) spin about its own axis (ii) move up and down while oscillating (iii)revolve in an elliptic path around its mean position.
Keep the pendulum oscillating for a few minutes. After the completion of few oscillations, start the stopwatch as the thread attached to the bob crosses the mean position. Consider it as a zero oscillation.
Keep counting the oscillation 1,2,3… n every time the bob crosses the mean position. Stop the stopwatch at the count of n oscillations. For better results, n should be chosen such that the time taken to complete n oscillations is 50 s or more. Read the total time taken for n oscillations. Repeat the observation a few times by noting down the time for the same n number of oscillations. Once noted down, take the mean of the readings. Calculate the time for one oscillation, i.e., the time period T ( = t / n ) of the pendulum.
Change the length of the pendulum, by about 10 cm. Repeat step 6 again for finding the time (t) for about 20 oscillations or more for the new length and find the mean time period. Take 5 or 6 more observations for different lengths of the pendulum and find the mean time period in each case.
Report observations in tabular form with proper units and significant figures.
Take effective length L along the x-axis and T 2 (or T ) along the y-axis, using the observed values from the table. Choose suitable scales on these axes to represent L and T 2 (or T ). Plot a graph between L and T 2 as shown in figure 2 and also between L and T as shown in figure 1.

Observation

The radius of the pendulum of the bob = ….. cm

Length of the hook = ….. cm

Least count of the meter scale = ….. mm

Least count of the stopwatch = ….. s

Plotting Graph

(i) L vs T Graph

Plot a graph between L versus T from observations recorded in the table above, taking L along x-axis and T along the y-axis. You will find that this graph is a curve, which is part of a parabola as shown in Figure 1.

(ii) L vs T 2 Graph

Plot a graph between L versus T 2 from observations recorded in the table, taking L along the x-axis and T 2 along the y-axis. You will find that the graph is a straight line passing through the origin as shown in figure 2.

(iii) From the L versus T 2 graph, determine the effective length of the second’s pendulum for T 2 = 4s 2 .

The graph L versus T is curved, convex upwards.

The graph L versus T 2 is a straight line.

The effective length of the second’s pendulum from the L versus T 2 graph is … cm.

Define simple pendulum.

Answer: A pendulum is defined as a single isolated particle suspended by a weightless Flexible and inextensible string with frictionless support.

Why is the word ‘simple’ used in simple pendulum?

Answer: Because the simple pendulum consists of mass m hanging from a string of length l and fixed at a pivot point P. The pendulums used in the wall clocks are known as a compound pendulum, and they have a metallic string instead of a thread.

Define simple harmonic motion (S.H.M).

Answer: A motion is called simple harmonic motion when:

(i)the magnitude of its acceleration is directly proportional to the displacement x from the mean position. (ii) the direction of the acceleration is always towards the mean position.

What is the relationship between frequency and time period?

Answer: The relationship between frequency and time period is given by f = 1/ T

What is restoring force?

Answer: The force that brings a vibrating body towards the mean position is known as the restoring force.

Can we use a cricket ball in the place of the bob?

Answer: No, we cannot. The bob must be as small as possible.

Can we use a rubber band instead of a thread?

Answer: No, we cannot because it is not inextensible.

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