1. Aim

To study the variation of time period of a simple pendulum of a given length by taking bobs of same size but different masses and interpret the result.

2. Apparatus Used

• A rigid support with clamp
• Inextensible thread
• Set of spherical bobs of same size but different masses (e.g., 50g, 100g, 150g, 200g, 250g)
• Stop watch (least count 0.01 s)
• Meter scale (least count 0.1 cm)
• Vernier caliper (least count 0.01 cm)
• Digital weighing scale

3. Diagram

4. Theory

A simple pendulum consists of a point mass (bob) suspended by a light inextensible string from a rigid support. When the bob is displaced from its equilibrium position and released, it oscillates about the equilibrium position.

When the bob is displaced through a small angle and released, it executes simple harmonic motion. The restoring force is provided by the component of gravitational force acting along the tangent to the arc.

For small angular displacements (θ < 5°), the time period of a simple pendulum is given by:

Where:

• \(T\) = Time period of the pendulum
• \(L\) = Length of the pendulum (from the point of suspension to the center of the bob)
• \(g\) = Acceleration due to gravity at that place

According to the formula, the time period of a simple pendulum:

• Is directly proportional to the square root of its length
• Is inversely proportional to the square root of acceleration due to gravity
• Is independent of the mass of the bob

This experiment aims to verify the last property - that the time period is independent of the mass of the bob when all other factors remain constant.

5. Formula

6. Procedure

1. Set up the apparatus as shown in the diagram with the rigid support and clamp.
2. Measure the mass of each bob using the digital weighing scale and record the values.
3. Attach a bob to one end of the thread and fix the other end to the clamp.
4. Measure the length (L) of the pendulum from the point of suspension to the center of the bob using a meter scale.
5. Displace the bob slightly (not more than 5° from the vertical) and release it to oscillate freely.
6. Start the stopwatch when the bob passes through the mean position and count "zero".
7. Count the number of oscillations (n = 20) and note the time taken for 20 complete oscillations.
8. Repeat step 7 three times for the same bob and calculate the average time.
9. Calculate the time period by dividing the average time by the number of oscillations.
10. Repeat steps 3-9 for each bob of different mass, keeping the length constant.
11. Record all the observations and calculations in the observation table.
12. Plot a graph between mass of the bob (x-axis) and time period (y-axis).

7. Observation Table

Length of the pendulum (L) = __________ cm

Number of oscillations (n) = 20

8. Calculations

For each bob of different mass:

1. Calculate the average time for 20 oscillations:

2. Calculate the time period:

3. Theoretical time period:

where g = 9.8 m/s²

4. Calculate the percentage error:

9. Result

1. From the observations and graph, it is observed that the time period of the simple pendulum remains approximately constant despite changing the mass of the bob.

2. This verifies that the time period of a simple pendulum is independent of the mass of the bob as predicted by the formula \(T = 2\pi\sqrt{\frac{L}{g}}\).

3. The average experimental time period = __________ s

4. The theoretical time period = __________ s

5. Percentage error = __________ %

10. Precautions

1. The amplitude of oscillations should be kept small (less than 5° from the vertical) to ensure simple harmonic motion.
2. The pendulum should oscillate freely without any external interference.
3. The thread used should be inextensible and of negligible mass compared to the bob.
4. The length of the pendulum should be measured accurately from the point of suspension to the center of the bob.
5. The stopwatch should be started when the bob passes through the mean position (equilibrium position).
6. The bob should not rotate about the vertical axis during oscillation.
7. Air resistance and air currents should be minimized as much as possible.
8. The support should be rigid and stable to avoid any vibration during the experiment.

11. Sources of Error

1. The suspension point might not be perfectly rigid, leading to energy dissipation.
2. Air resistance can affect the oscillations, especially for lighter bobs.
3. The string might have some mass, which is not accounted for in the theoretical formula.
4. Human reaction time in starting and stopping the stopwatch might introduce timing errors.
5. Measuring the length of the pendulum inexactly, particularly determining the center of the bob.
6. If the angle of displacement exceeds 5°, the small angle approximation (sin θ ≈ θ) becomes less accurate.
7. Temperature variations may cause changes in the length of the thread.
8. Slight rotational motion of the bob can affect the time period.

12. Viva Voice Questions

1. Q: Why is the time period of a simple pendulum independent of the mass of the bob?

   A: The gravitational force acting on the bob is proportional to its mass, and the inertia of the bob is also proportional to its mass. According to Newton's Second Law, these two effects cancel out, making the time period independent of mass.

2. Q: What happens to the time period if the length of the pendulum is doubled?

   A: If the length is doubled, the time period increases by a factor of √2 (approximately 1.414 times) because \(T \propto \sqrt{L}\).

3. Q: Does the time period of a simple pendulum depend on the amplitude of oscillation?

   A: For small amplitudes (less than 5°), the time period is approximately independent of amplitude. For larger amplitudes, the time period increases slightly with increasing amplitude.

4. Q: How would the time period change if this experiment is performed on the moon?

   A: On the moon, the acceleration due to gravity is about 1/6th of that on Earth. Since \(T \propto 1/\sqrt{g}\), the time period would increase by a factor of √6 (approximately 2.45 times).

5. Q: What is a simple harmonic motion?

   A: Simple harmonic motion is a type of oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction to the displacement.

6. Q: Why do we consider only small angular displacements in this experiment?

   A: For small angles, the approximation sin θ ≈ θ holds true, which makes the motion simple harmonic. For larger angles, this approximation fails, and the pendulum doesn't execute perfect simple harmonic motion.

7. Q: How can you determine the value of 'g' using a simple pendulum?

   A: By measuring the length (L) and time period (T) of the pendulum, we can calculate g using the formula: \(g = \frac{4\pi^2 L}{T^2}\).