Determination of Acceleration Due to Gravity Using Simple Pendulum

1. Aim

To determine the acceleration due to gravity (g) at a place by using a simple pendulum and to study the relationship between the length of the pendulum and its time period.

2. Apparatus Used

• Small heavy metallic bob
• Fine, inextensible string (about 1-1.5m long)
• Retort stand with clamp
• Stopwatch (least count 0.1s)
• Meter scale (least count 1mm)
• Vernier calipers (to measure bob diameter)
• Split cork or knife edge for suspension

3. Diagram

Figure: Simple pendulum showing length (l), bob diameter (d), and angular displacement (θ)

4. Theory

A simple pendulum consists of a point mass suspended by a weightless, inextensible string from a rigid support. When displaced from its mean position and released, it executes simple harmonic motion (for small angles θ ≤ 5°).

The time period (T) of oscillation of a simple pendulum is given by:

Where:

• \( T \) = Time period of oscillation
• \( L \) = Effective length of pendulum (string length + radius of bob)
• \( g \) = Acceleration due to gravity

Rearranging the formula gives the expression for g:

For small oscillations (θ ≤ 5°), the time period is independent of:

• Mass of the bob
• Amplitude of oscillation

5. Formula

The acceleration due to gravity is calculated using:

Where:

• Effective length \( L = l + r \)
  \[ l = \text{length of string} \] \[ r = \text{radius of bob} = \frac{d}{2} \]
• Time period \( T = \frac{\text{Time for n oscillations}}{n} \)

6. Procedure

1. Measure the diameter (d) of the bob using vernier calipers and calculate radius (r = d/2).
2. Set up the pendulum by suspending the bob with the string from the rigid support.
3. Adjust the length (l) of the string to about 50 cm (from point of suspension to top of bob).
4. Measure the exact length (l) using the meter scale.
5. Calculate effective length: \( L = l + r \).
6. Displace the bob slightly (≤ 5°) and release to start oscillations.
7. Measure time for 20 oscillations using stopwatch and calculate time period (T).
8. Repeat the measurement for the same length 3 times and take average T.
9. Repeat steps 3-8 for different lengths (e.g., 60cm, 70cm,..., 100cm).
10. Record all observations in the tabular form.
11. Plot graph of \( T^2 \) vs L and determine g from the slope.

7. Observation Table

Diameter of bob (d) = _____ cm

Radius of bob (r = d/2) = _____ cm

8. Calculations

1. For each length, calculate effective length:
   \[ L = l + r \]
2. Calculate mean time period for each length:
   \[ T = \frac{\text{Mean time for 20 oscillations}}{20} \]
3. Calculate \( T^2 \) for each length.
4. Plot graph of \( T^2 \) (y-axis) vs L (x-axis). The graph should be a straight line.
5. Determine slope (m) of the graph:
   \[ m = \frac{\Delta T^2}{\Delta L} \]
6. Calculate g using:
   \[ g = 4\pi^2 \frac{1}{m} \]
7. Alternatively, calculate g for each length using:
   \[ g = 4\pi^2 \frac{L}{T^2} \]
   and take average.

9. Result

• The acceleration due to gravity (g) determined from the experiment = _____ m/s²
• Standard value of g at the location = _____ m/s²
• Percentage error = _____ %
• The graph of \( T^2 \) vs L is a straight line, verifying \( T \propto \sqrt{L} \) relationship

10. Precautions

• Ensure the string is inextensible and the support is rigid
• Amplitude of oscillation should be small (≤ 5°)
• Time at least 20 oscillations to reduce timing errors
• Measure length from point of suspension to center of bob
• Avoid air currents that may affect oscillations
• Repeat measurements for each length to minimize errors
• Start and stop stopwatch at the mean position
• Ensure oscillations are in one plane without rotation

11. Viva Voce Questions