Bar Pendulum Experiment

Determination of Acceleration Due to Gravity Using Bar Pendulum

1. Aim

To determine the acceleration due to gravity (g) at a place by using a bar pendulum and to study the variation of time period with the distance of the point of suspension from the center of gravity.

2. Apparatus Used

  • Bar pendulum (metallic bar with holes)
  • Knife-edge support
  • Stopwatch (least count 0.1s)
  • Meter scale (least count 1mm)
  • Spirit level
  • Graph paper

3. Diagram

Figure: Bar pendulum suspended from knife edge

4. Theory

A bar pendulum is a rigid body of definite shape that can oscillate about a horizontal axis passing through it. When displaced from its mean position, it executes angular simple harmonic motion.

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

\[ T = 2\pi \sqrt{\frac{I}{mgh}} \]

Where:

  • \( T \) = Time period of oscillation
  • \( I \) = Moment of inertia about the axis of suspension
  • \( m \) = Mass of the pendulum
  • \( g \) = Acceleration due to gravity
  • \( h \) = Distance between the point of suspension and the center of gravity

Using the parallel axis theorem (\( I = I_0 + mh^2 \)), where \( I_0 \) is the moment of inertia about the center of gravity, the expression becomes:

\[ T = 2\pi \sqrt{\frac{I_0 + mh^2}{mgh}} \]

For a bar pendulum of length \( L \), \( I_0 = \frac{mL^2}{12} \), giving:

\[ T = 2\pi \sqrt{\frac{L^2 + 12h^2}{12gh}} \]

5. Formula

The acceleration due to gravity can be calculated using:

\[ g = 4\pi^2 \frac{L^2 + 12h^2}{12hT^2} \]

At the point where the time period is minimum (\( h = \frac{L}{\sqrt{12}} \)), the formula simplifies to:

\[ g = \frac{4\pi^2 L}{T_{min}^2 \sqrt{3}} \]

6. Procedure

  1. Determine the center of gravity (CG) of the bar by balancing it on a knife edge.
  2. Measure the length (L) of the bar from end to end.
  3. Suspend the bar from one of the holes at a distance h from the CG.
  4. Displace the bar slightly (about 5°) and release to start oscillations.
  5. Measure the time for 20 oscillations and calculate the time period (T).
  6. Repeat the measurement for different suspension points on both sides of the CG.
  7. Record observations in a tabular form.
  8. Plot a graph of T versus h and determine \( T_{min} \).
  9. Calculate 'g' using the appropriate formula.

7. Observation Table

S.No. Distance from CG (h) in cm Time for 20 oscillations (s) Mean Time Period (T) in s
Trial 1 Trial 2 Trial 3
1 h1
2 h2
... ...

Length of bar (L) = _____ cm

Mass of bar (m) = _____ g

8. Calculations

  1. For each h, calculate the mean time period T using:
    \[ T = \frac{\text{Time for 20 oscillations}}{20} \]
  2. Plot a graph of T (y-axis) versus h (x-axis). The curve will be symmetrical about the CG.
  3. From the graph, determine \( T_{min} \) and corresponding \( h_{min} \).
  4. Calculate g using:
    \[ g = 4\pi^2 \frac{L^2 + 12h^2}{12hT^2} \]
    for different h values and take average.
  5. Alternatively, use:
    \[ g = \frac{4\pi^2 L}{T_{min}^2 \sqrt{3}} \]
    with \( T_{min} \) from the graph.

Graph Plotting Instructions

Plot T (y-axis) vs h (x-axis):

  • The curve should show two symmetrical branches
  • Mark the minimum time period point (\( T_{min} \))
  • Draw horizontal lines corresponding to \( T_1 = T_2 \) to find equivalent length points

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 = _____ %

10. Precautions

  • Ensure the knife edge is horizontal using a spirit level
  • Amplitude of oscillation should be small (≤ 5°)
  • Time at least 20 oscillations to reduce timing errors
  • Measure distances from CG accurately
  • Avoid air currents that may affect oscillations
  • Repeat measurements for each position to minimize errors

11. Viva Voce Questions

1. What is a compound pendulum? How does it differ from a simple pendulum?
2. Derive the expression for the time period of a bar pendulum.
3. Why does the time period have a minimum value in this experiment?
4. What is the significance of the center of gravity in this experiment?
5. How would you determine the radius of gyration of the bar pendulum?
6. What are the sources of error in this experiment?
7. How does the time period change if the bar is suspended closer to its ends?
8. What would be the shape of the T vs h graph? Why?
9. How can you verify that \( h_1h_2 = k^2 \) where k is the radius of gyration?
10. What practical applications does this experiment have?
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