To Study Damping of a Bar Pendulum
(Logarithmic Decrement)

1. Aim

To study the damped oscillations of a bar pendulum and determine the damping coefficient and logarithmic decrement.

2. Apparatus Used

• Bar pendulum with knife edge
• Stopwatch/timer
• Meter scale
• Vernier caliper
• Weight clamp
• Optical/mechanical amplitude measurement setup
• Graph paper
• Calculator

3. Diagram

4. Theory

A bar pendulum consists of a uniform metal bar suspended from a knife edge. When displaced from its equilibrium position and released, it executes oscillations. In an ideal system (without damping), these oscillations would continue indefinitely with constant amplitude. However, in real systems, various damping forces such as air resistance and friction at the pivot cause the amplitude to decrease with time.

The equation of motion for a damped oscillator is:

Where:

• $\theta$ is the angular displacement
• $\beta$ is the damping coefficient
• $\omega_0$ is the natural angular frequency of the undamped oscillator

The solution to this equation for an underdamped system ($\beta < \omega_0$) is:

Where:

• $A_0$ is the initial amplitude
• $\omega_d = \sqrt{\omega_0^2 - \beta^2}$ is the damped angular frequency
• $\phi$ is the phase angle

The amplitude of oscillation decreases exponentially with time:

The logarithmic decrement ($\delta$) is defined as the natural logarithm of the ratio of two consecutive amplitudes:

Where T is the time period of oscillation. By measuring the amplitude of successive oscillations, we can determine the logarithmic decrement and hence the damping coefficient.

5. Formula

1. Time period of oscillation:

   $$T = \frac{\text{Time for n oscillations}}{n}$$

2. Logarithmic decrement:

   $$\delta = \frac{1}{m} \ln\left(\frac{A_1}{A_{m+1}}\right)$$

   Where m is the number of oscillations between the measured amplitudes $A_1$ and $A_{m+1}$

3. Damping coefficient:

   $$\beta = \frac{\delta}{T}$$

4. Quality factor:

   $$Q = \frac{\pi}{\delta}$$

6. Procedure

1. Set up the bar pendulum on a rigid support with the knife edge.
2. Adjust the position of the bar so that it hangs vertically in equilibrium.
3. Mark a reference point directly below the bar to indicate the equilibrium position.
4. Measure the length of the bar pendulum using a meter scale.
5. Gently displace the pendulum from its equilibrium position by a small angle (≤ 5°).
6. Release the pendulum (without giving it any initial velocity) and start the stopwatch simultaneously.
7. Count the oscillations and record the time for 20 complete oscillations to calculate the time period.
8. To measure the damping:
   1. Set up a measurement system to record the amplitude of oscillations.
   2. Displace the pendulum by a small angle and release it.
   3. Record the amplitudes of successive oscillations ($A_1$, $A_2$, $A_3$, ..., $A_n$).
   4. Repeat this procedure 3 times for better accuracy.
9. Plot a graph of ln($A_n$) vs. number of oscillations (n). The slope of this graph gives the logarithmic decrement.

7. Observation Table

Table 1: Measurement of Time Period

Observation	Number of oscillations (n)	Time taken (s)	Time period T = time/n (s)
1	20
2	20
3	20
Mean	-	-

Table 2: Measurement of Amplitude

Oscillation number (n)	Amplitude ($A_n$) (cm)	ln($A_n$)
1
2
3
4
5
6
7
8
9
10

8. Calculations

1. Calculate the mean time period from Table 1:

   $$T_{mean} = \frac{T_1 + T_2 + T_3}{3}$$

2. Calculate the logarithmic decrement using:

   $$\delta = \frac{1}{m} \ln\left(\frac{A_1}{A_{m+1}}\right)$$

   OR from the slope of the ln($A_n$) vs. n graph

3. Calculate the damping coefficient:

   $$\beta = \frac{\delta}{T_{mean}}$$

4. Calculate the quality factor:

   $$Q = \frac{\pi}{\delta}$$

9. Graph

Plot a graph with:

• X-axis: Number of oscillations (n)
• Y-axis: Natural logarithm of amplitude ln($A_n$)
• The slope of this graph will be equal to the negative of the logarithmic decrement (-δ)

10. Result

1. The time period of the bar pendulum: _____ seconds
2. Logarithmic decrement (δ): _____
3. Damping coefficient (β): _____ s⁻¹
4. Quality factor (Q): _____

11. Precautions

1. The support for the knife edge should be rigid and free from vibrations.
2. The knife edge should be sharp and horizontal.
3. Initial displacement should be small (≤ 5°) to ensure simple harmonic motion.
4. The pendulum should be released gently without giving it any initial velocity.
5. The bar should be symmetric about its center of mass.
6. Air currents should be minimized in the laboratory during the experiment.
7. Time measurement should begin when the pendulum passes through its equilibrium position.
8. For amplitude measurements, ensure the reference point is properly marked.
9. Take multiple readings to minimize random errors.
10. The bar pendulum should oscillate in a single plane without any twisting motion.

12. Viva Voce Questions

1. What is a bar pendulum? How does it differ from a simple pendulum?
2. Define damping and explain different types of damping.
3. What causes damping in a bar pendulum?
4. Define logarithmic decrement. What is its physical significance?
5. How does the time period of a damped oscillator differ from that of an undamped oscillator?
6. What is the quality factor (Q)? What does a high Q-value indicate?
7. How would increasing the air resistance affect the logarithmic decrement?
8. What would happen if the damping coefficient exceeds the natural frequency?
9. Explain the difference between forced and free oscillations.
10. How does the position of the center of oscillation affect the time period of a bar pendulum?
11. What is the relationship between damping coefficient and logarithmic decrement?
12. How would you experimentally verify that the amplitude decreases exponentially with time?
13. Why is it necessary to keep the initial amplitude small in this experiment?
14. How would adding weights at different positions on the bar affect the damping?
15. Explain how energy dissipation occurs in a damped oscillatory system.