To Study Dissipation of Energy of a Simple Pendulum
Aim of the Experiment
To study the dissipation of energy of a simple pendulum by plotting a graph between square of amplitude (A²) and time (t), and to analyze the damping effect in oscillatory motion.
Apparatus Required
- A simple pendulum (a small metallic bob attached to a light, inextensible string)
- A rigid support with clamp
- A stopwatch/timer
- A meter scale
- A protractor or angle measuring device
- Graph paper
- Calculator
Theory
A simple pendulum consists of a point mass (bob) suspended by a massless, inextensible string from a rigid support. When displaced from its equilibrium position and released, the pendulum oscillates about its mean position.
The time period of a simple pendulum is given by:
$$T = 2\pi\sqrt{\frac{L}{g}}$$
where:
- $T$ = Time period of the pendulum
- $L$ = Length of the pendulum
- $g$ = Acceleration due to gravity
In an ideal scenario, a simple pendulum would oscillate indefinitely with constant amplitude. However, in practical situations, the amplitude gradually decreases with time due to various dissipative forces like air resistance and friction at the support.
The energy of the pendulum at any instant is proportional to the square of its amplitude:
$$E \propto A^2$$
where $A$ is the amplitude of oscillation.
For small damping, the amplitude decreases exponentially with time:
$$A = A_0 e^{-\beta t}$$
where:
- $A_0$ = Initial amplitude
- $\beta$ = Damping coefficient
- $t$ = Time elapsed
Squaring both sides:
$$A^2 = A_0^2 e^{-2\beta t}$$
Taking natural logarithm on both sides:
$$\ln(A^2) = \ln(A_0^2) - 2\beta t$$
This equation represents a straight line when $\ln(A^2)$ is plotted against time $t$, with slope $-2\beta$ and y-intercept $\ln(A_0^2)$. The damping coefficient $\beta$ can be determined from the slope of this line.
Experimental Setup
Figure 1: Experimental setup of a simple pendulum showing the position of equilibrium and extreme positions.
Procedure
Set up the simple pendulum by attaching the bob to the string and fixing the other end to the rigid support using the clamp. Ensure that the bob can swing freely without any obstruction.
Measure the length of the pendulum from the point of suspension to the center of the bob using a meter scale. Record this value as $L$.
Displace the bob from its equilibrium position by a small angle (preferably less than 15° to ensure simple harmonic motion) and measure this initial angular displacement $\theta_0$.
Release the bob from rest and start the stopwatch simultaneously. Let the pendulum oscillate freely.
At regular time intervals (e.g., every 30 seconds or after every 10 oscillations), record the amplitude of oscillation by observing the maximum angular displacement $\theta$ of the pendulum from its equilibrium position.
Continue the observations for about 10-15 minutes or until the amplitude becomes too small to measure accurately.
Calculate the linear amplitude $A$ corresponding to each angular displacement using the formula $A = L\sin\theta$ or $A = L\theta$ (for small angles in radians).
Calculate the square of amplitude $A^2$ for each observation.
Plot a graph of $A^2$ versus time $t$. This should be an exponentially decreasing curve.
To determine the damping coefficient, plot a graph of $\ln(A^2)$ versus time $t$. This should give a straight line with negative slope.
Calculate the slope of this straight line and determine the damping coefficient $\beta$ using the relation: Slope = $-2\beta$.
Observations
Length of the pendulum, $L$ = ______________ m
Initial angular displacement, $\theta_0$ = ______________ degrees
Initial amplitude, $A_0 = L\sin\theta_0$ = ______________ m
| S.No. | Time, $t$ (s) | Angular Displacement, $\theta$ (degrees) | Amplitude, $A = L\sin\theta$ (m) | Square of Amplitude, $A^2$ (m²) | Natural Log of $A^2$, $\ln(A^2)$ |
|---|---|---|---|---|---|
| 1 | 0 | ||||
| 2 | 30 | ||||
| 3 | 60 | ||||
| 4 | 90 | ||||
| 5 | 120 | ||||
| 6 | 150 | ||||
| 7 | 180 | ||||
| 8 | 210 | ||||
| 9 | 240 | ||||
| 10 | 270 |
Graphs
Graph 1: Square of Amplitude ($A^2$) vs Time ($t$)
Insert your graph here
Graph 2: Natural Logarithm of Square of Amplitude ($\ln(A^2)$) vs Time ($t$)
Insert your graph here
Calculations
Slope of the line in the graph of $\ln(A^2)$ vs $t$ = ______________ s⁻¹
Damping coefficient, $\beta = -\frac{\text{Slope}}{2}$ = ______________ s⁻¹
The time constant, $\tau = \frac{1}{\beta}$ = ______________ s
Time taken for the energy to reduce to half its initial value, $t_{1/2} = \frac{\ln 2}{\beta}$ = ______________ s
Precautions
- Ensure that the pendulum oscillates in a single vertical plane without any lateral movements.
- The angular displacement should be kept small (preferably less than 15°) to ensure that the motion is simple harmonic.
- The string of the pendulum should be light, inextensible, and as thin as possible.
- Avoid any air drafts in the vicinity of the pendulum during the experiment.
- Take readings quickly and accurately to minimize errors.
- The pendulum bob should be released gently without imparting any initial velocity.
- The support should be rigid to avoid any energy loss due to vibrations.
Sources of Error
- Air resistance causing damping of the oscillations
- Friction at the point of suspension
- Non-ideal nature of the string (having some mass and extensibility)
- Errors in measuring the angular displacement
- Human reaction time errors in the time measurement
- The pendulum may not oscillate in a perfect vertical plane
- The bob may not be a perfect point mass
Conclusion
From this experiment, we have studied the dissipation of energy in a simple pendulum due to damping forces. We observed that:
- The square of the amplitude ($A^2$) decreases exponentially with time, confirming that the energy of the pendulum (which is proportional to $A^2$) also decreases exponentially.
- The damping coefficient ($\beta$) of the pendulum system is found to be __________ s⁻¹.
- The time constant of the system is __________ s, indicating the time taken for the amplitude to reduce to 1/e of its initial value.
- The energy of the pendulum reduces to half its initial value in approximately __________ s.
This experiment demonstrates how friction and air resistance gradually dissipate the mechanical energy of an oscillating system, causing its amplitude to decrease over time.
Discussions
Click the button below to reveal explanations about the energy dissipation in a simple pendulum:
Print Instructions
To print this worksheet:
- Fill in all required measurements and observations in the tables.
- Insert the required graphs in the designated areas.
- Complete all calculations in the provided spaces.
- Click the "Print Worksheet" button below.