Using a Simple Pendulum, Plot L-T² Graph and Find Effective Length of Second's Pendulum
1. Aim
To determine the relationship between the length (L) of a simple pendulum and the square of its time period (T²), and to find the effective length of a second's pendulum.
2. Apparatus Used
- A pendulum bob (spherical metal bob)
- Thread or thin string
- Meter scale or measuring tape
- Laboratory stand with clamp
- Stopwatch/digital timer
- Graph paper
- Vernier caliper (to measure the diameter of the bob)
3. Diagram
Fig. 1: Experimental setup of a simple pendulum. The length L is measured from the point of suspension to the center of the bob.
4. Theory
A simple pendulum consists of a small heavy 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.
According to the laws of simple harmonic motion, the time period (T) of a simple pendulum is given by:
$T = 2\pi\sqrt{\frac{L}{g}}$
where:
- T = Time period of oscillation (in seconds)
- L = Length of the pendulum (in meters)
- g = Acceleration due to gravity (9.8 m/s²)
Squaring both sides of the equation:
$T^2 = \frac{4\pi^2}{g} \times L$
This equation shows that T² is directly proportional to L. When plotted on a graph, T² versus L should yield a straight line with slope = 4π²/g.
A second's pendulum is one whose time period is exactly 2 seconds (i.e., one complete oscillation takes 2 seconds). From the pendulum equation:
$L = \frac{T^2 \times g}{4\pi^2}$
For T = 2 seconds:
$L = \frac{4 \times 9.8}{4\pi^2} \approx \frac{39.2}{39.5} \approx 0.993 \text{ meters}$
Therefore, the theoretical length of a second's pendulum is approximately 0.993 meters or 99.3 cm.
5. Formula
The key formulas used in this experiment are:
1. Time period of a simple pendulum:
$T = 2\pi\sqrt{\frac{L}{g}}$2. Relationship between T² and L:
$T^2 = \frac{4\pi^2}{g} \times L$3. Slope of the T² vs L graph:
$\text{Slope} = \frac{4\pi^2}{g}$4. To find acceleration due to gravity:
$g = \frac{4\pi^2}{\text{Slope}}$5. Length of second's pendulum (when T = 2s):
$L = \frac{4 \times g}{4\pi^2}$6. Procedure
- Set up the pendulum by attaching the bob to the thread and securing the other end to the laboratory stand.
- Measure the initial length (L) of the pendulum from the point of suspension to the center of the bob using the meter scale.
- Displace the bob slightly (about 5° from the vertical) and release it to start oscillations.
- Start the stopwatch when the bob passes its mean position and count 20 complete oscillations.
- Record the time taken for 20 oscillations and divide by 20 to get the time period (T).
- Repeat steps 3-5 three times for the same length to obtain an average value of T.
- Calculate T² for this length.
- Change the length of the pendulum and repeat steps 2-7 for at least 6 different lengths.
- Record all observations in the observation table.
- Plot a graph of T² (y-axis) versus L (x-axis).
- Determine the slope of the graph.
- Use the slope to calculate the value of g.
- Calculate the length of the second's pendulum using the formula.
7. Observation Table
| S.No. | Length of Pendulum (L) in cm | Time for 20 oscillations (in seconds) | Time Period (T = time/20) in seconds | T² (s²) |
|---|---|---|---|---|
| 1 | 40 | 25.6 | 1.28 | 1.64 |
| 2 | 60 | 31.4 | 1.57 | 2.46 |
| 3 | 80 | 36.2 | 1.81 | 3.28 |
| 4 | 100 | 40.4 | 2.02 | 4.08 |
| 5 | 120 | 44.2 | 2.21 | 4.88 |
| 6 | 140 | 47.8 | 2.39 | 5.71 |
8. Calculations
Step 1: Plot a graph of T² versus L. The graph should be a straight line passing through the origin.
Step 2: Calculate the slope of the graph.
$\text{Slope} = \frac{\text{Change in } T^2}{\text{Change in } L} = \frac{5.71 - 1.64}{140 - 40} = \frac{4.07}{100} = 0.0407 \text{ s²/cm}$
Step 3: Convert the slope to SI units (s²/m).
$\text{Slope in SI} = 0.0407 \text{ s²/cm} \times 100 = 4.07 \text{ s²/m}$
Step 4: Calculate the experimental value of g.
$g = \frac{4\pi^2}{\text{Slope}} = \frac{4 \times 9.87}{4.07} = \frac{39.48}{4.07} = 9.70 \text{ m/s²}$
Step 5: Calculate the length of the second's pendulum (when T = 2s).
$L = \frac{T^2 \times g}{4\pi^2} = \frac{4 \times 9.70}{4\pi^2} = \frac{38.8}{39.48} = 0.983 \text{ m} = 98.3 \text{ cm}$
Step 6: Calculate the percentage error.
$\text{Percentage Error in } g = \frac{|9.8 - 9.7|}{9.8} \times 100\% = 1.02\%$
9. Result
- The graph between length (L) and time period squared (T²) is a straight line, confirming that T² ∝ L.
- Experimental value of acceleration due to gravity (g) = 9.70 m/s²
- Theoretical value of g = 9.8 m/s²
- Percentage error = 1.02%
- The effective length of the second's pendulum is found to be 98.3 cm.
10. Precautions
- The amplitude of oscillation should be kept small (< 5°) to ensure simple harmonic motion.
- The pendulum should be released gently without giving it any push.
- The thread should be light, inextensible, and straight without any kinks.
- The bob should be heavy and spherical to minimize air resistance.
- The time measurement should start when the bob passes through the mean position.
- A large number of oscillations (at least 20) should be timed to minimize reaction time errors.
- The point of suspension should be rigid and free from vibrations.
- The length should be measured from the point of suspension to the center of the bob.
- Oscillations should be in a single vertical plane to avoid any pendulum motion.
11. Sources of Error
- Measurement errors: Errors in measuring the length of the pendulum or the time period.
- Air resistance: Can affect the oscillations, especially for lighter bobs.
- Non-ideal string: If the string has significant mass or is not inextensible.
- Large amplitude: For large amplitudes, the simple pendulum equation is approximate.
- Effective length determination: The effective length includes part of the suspension system and is not just from the point of suspension to the center of the bob.
- Human reaction time: Introduces errors in starting and stopping the stopwatch.
- Damping: Gradual decrease in amplitude due to friction and air resistance.
- Non-uniform gravitational field: Variations in g due to geographical location.
- Temperature effects: Changes in length of the thread due to temperature variations.
12. Viva Voice Questions
Q1. What is a simple pendulum?
A simple pendulum is an idealized model consisting of a point mass (bob) suspended from a fixed point by a massless, inextensible string, and subject only to the forces of gravity and tension in the string.
Q2. Why must the amplitude of oscillation be kept small?
For small amplitudes (< 5°), the motion of a simple pendulum can be approximated as simple harmonic motion, where the period is independent of amplitude. At larger angles, this approximation breaks down, and the pendulum no longer follows the simple T = 2π√(L/g) relationship accurately.
Q3. What is a second's pendulum?
A second's pendulum is one whose time period is exactly 2 seconds, meaning it takes 2 seconds to complete one full oscillation (from one extreme to the other and back). Its theoretical length at standard gravity is approximately 99.3 cm.
Q4. How does the length of a pendulum affect its time period?
The time period (T) of a pendulum is directly proportional to the square root of its length (L). Mathematically, T = 2π√(L/g). So, if the length is increased by a factor of 4, the time period will increase by a factor of 2.
Q5. Why is the graph between T² and L a straight line?
Since T = 2π√(L/g), we get T² = (4π²/g)L. This is in the form y = mx, which is the equation of a straight line passing through the origin, where the slope m = 4π²/g.
Q6. How would the time period of a pendulum change on the moon?
The gravitational acceleration on the moon is about 1/6th that of Earth. Since T = 2π√(L/g), and T is inversely proportional to √g, the time period on the moon would be √6 (approximately 2.45) times longer than on Earth for the same pendulum length.
Q7. Does the mass of the bob affect the time period?
No, the time period of a simple pendulum is independent of the mass of the bob. This is because while the gravitational force increases with mass, so does the inertia, and these effects cancel out in the equation of motion.
Q8. What happens to the time period if the pendulum is taken to a higher altitude?
The value of g decreases with increasing altitude, which would cause the time period to increase slightly as T ∝ 1/√g.
Q9. Why do we measure the time for multiple oscillations instead of just one?
Measuring multiple oscillations and then calculating the average time period helps minimize human reaction time errors and provides a more accurate measurement of the time period.
Q10. What is the effective length of a pendulum with a large spherical bob?
The effective length is the distance from the point of suspension to the center of oscillation. For a spherical bob, this is slightly different from the distance to the center of mass due to the bob's moment of inertia. If the bob's radius is much smaller than the string length, this difference is usually negligible.