Class 11 Physics Notes Chapter 13 (Chapter 13) – Lab Manual (English) Book

Detailed Notes with MCQs of Experiment number 13 from your Physics Lab Manual. This experiment deals with the Simple Pendulum and is quite important, not just for your practical exams but also because concepts related to it often appear in various government competitive exams.

We'll be looking at determining the relationship between the length of a simple pendulum and its time period, and using that relationship to find 'g' (acceleration due to gravity) or the effective length of a second's pendulum.

Experiment 13: Simple Pendulum - Time Period and Length Relationship

1. Aim:
To study the relationship between the time period (T) and length (L) of a simple pendulum and to determine the effective length of a second's pendulum using a graph. (Alternatively: To determine the value of 'g' using a simple pendulum).

2. Apparatus Required:

• A heavy metallic sphere (bob) with a hook.
• A light, strong, inextensible cotton thread.
• A rigid support (clamp stand).
• A stopwatch (preferably with a least count of 0.1s or 0.01s).
• A metre scale.
• Split cork.
• Vernier callipers (optional, for measuring the diameter of the bob accurately).

3. Theory:

• Simple Pendulum: An ideal simple pendulum consists of a point mass (bob) suspended by a weightless, inextensible, and perfectly flexible string from a rigid support about which it can oscillate freely. In practice, we use a small, heavy metallic sphere suspended by a light cotton thread.
• Oscillation: One complete to-and-fro movement of the pendulum bob about its mean position is called one oscillation.
• Time Period (T): The time taken by the pendulum to complete one oscillation is called its time period.
• Frequency (f): The number of oscillations completed per second (f = 1/T). Unit: Hertz (Hz).
• Length of the Pendulum (L): The effective length of a simple pendulum is the distance from the point of suspension to the centre of gravity of the bob.
  • If 'l' is the length of the thread from the point of suspension to the hook, 'h' is the length of the hook, and 'r' is the radius of the bob, then the effective length L = l + h + r. Often, 'h' is negligible or included in 'l', so L ≈ l + r.
• Formula for Time Period: For small angular displacements (amplitude < 15°), the time period (T) of a simple pendulum is given by:
  T = 2π √(L/g)
  Where:
  • T = Time Period (in seconds)
  • L = Effective length of the pendulum (in metres)
  • g = Acceleration due to gravity (in m/s²)
• Relationship between T and L: Squaring the formula, we get:
  T² = (4π²/g) * L
  This shows that T² ∝ L (since 4π²/g is constant at a given place).
  Therefore, a graph of L versus T² should be a straight line passing through the origin.
• Second's Pendulum: A pendulum whose time period is exactly two seconds (T = 2s) is called a second's pendulum. Its effective length (Ls) can be found by substituting T=2s in the formula:
  2 = 2π √(Ls/g)
1 = π √(Ls/g)
1 = π² (Ls/g)
Ls = g / π²
  The length of a second's pendulum is approximately 1 metre (≈ 99.4 cm) at places where g ≈ 9.8 m/s².

4. Procedure Outline:

• Measure the radius 'r' of the bob using Vernier callipers (or diameter/2).
• Tie the bob to the thread and suspend it from the rigid support using split corks.
• Adjust the length of the thread ('l') so that the effective length L = l + r is a desired value (e.g., 70 cm, 80 cm, 90 cm...). Measure 'l' carefully from the point of suspension to the top of the bob.
• Displace the bob slightly (small amplitude) and let it oscillate in a vertical plane.
• Start the stopwatch as the bob passes through its mean position and simultaneously start counting oscillations (start count from zero).
• Measure the time taken for 20 or 30 complete oscillations (t).
• Calculate the time period T = t / (number of oscillations).
• Repeat steps 3-7 for different lengths (L). Record at least 5-6 readings.
• Record observations in a tabular format.

5. Observations Table (Sample):
Radius of bob, r = ... cm = ... m
Least count of stopwatch = ... s
Least count of metre scale = ... cm = ... m

6. Calculations & Graph:

• Calculate T and T² for each length L.
• Plot Graph: Plot a graph with L (in metres) on the X-axis and T² (in s²) on the Y-axis. It should be a straight line passing through (or close to) the origin.
• Calculate Slope: Find the slope of the L vs T² graph: Slope = ΔT² / ΔL.
• Determine 'g': From the formula T² = (4π²/g) * L, we have T²/L = 4π²/g.
  The slope of the T² vs L graph is Slope = ΔT² / ΔL = 4π²/g.
  Therefore, g = 4π² / Slope. (If you plot L vs T², then Slope = ΔL / ΔT² = g / 4π², so g = 4π² * Slope). Be careful which axis represents which quantity! Conventionally, L is independent (X-axis) and T² is dependent (Y-axis).
• Determine Length of Second's Pendulum (Ls): On the L vs T² graph, find the value of L corresponding to T² = 4 s² (since T=2s for a second's pendulum). This value of L is Ls. Alternatively, use the calculated value of 'g' in Ls = g / π².

7. Result:

• The graph between L and T² is a straight line, showing T² ∝ L.
• The value of 'g' calculated from the graph is ... m/s².
• The effective length of the second's pendulum determined from the graph is ... m.

8. Precautions:

• The thread should be light, strong, and inextensible.
• The support must be rigid.
• The amplitude of oscillation should be small (bob should not swing too wide) so that the approximation sin θ ≈ θ holds, which is assumed in the derivation of the formula.
• The bob should oscillate in a vertical plane without spinning.
• The length of the pendulum should be measured accurately from the point of suspension to the centre of gravity of the bob.
• Measure the time for a large number of oscillations (e.g., 20 or 30) to minimize error in measuring the time period.
• Start and stop the stopwatch at the mean position, where the bob's speed is maximum, reducing reaction time error.
• Avoid air currents that might affect the oscillations.

9. Sources of Error:

• The support may not be perfectly rigid.
• The thread may not be perfectly weightless or inextensible.
• Amplitude may not be small.
• Air resistance affects the motion.
• Error in measuring length (parallax error, least count error).
• Error in measuring time (personal error/reaction time, least count error of stopwatch).
• The bob may spin or not move in a perfect vertical plane.
• Non-uniform density of the bob (Centre of gravity not exactly at the geometric centre).

10. Key Concepts for Exams (Viva/Theory):

• Definition of simple pendulum, oscillation, time period, frequency, effective length.
• Formula T = 2π√(L/g) and its derivation assumptions (small amplitude).
• Factors affecting T: Length (T ∝ √L), Acceleration due to gravity (T ∝ 1/√g).
• Factors not affecting T (ideally): Mass of the bob, Amplitude (for small angles).
• Concept of a second's pendulum and its approximate length.
• Interpretation of L vs T and L vs T² graphs. Shape of L vs T graph is a parabola opening towards the L-axis. Shape of L vs T² graph is a straight line through the origin.
• How 'g' can be determined from the slope of the graph.
• Why amplitude should be small.
• Why time for many oscillations is measured.
• Effect of taking the pendulum to the moon (g decreases, T increases) or into a mine (g decreases, T increases) or to a mountain top (g decreases, T increases).

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the simple pendulum experiment:

1. The time period of a simple pendulum depends on its:
   a) Mass of the bob
   b) Amplitude of oscillation
   c) Length of the pendulum
   d) Material of the bob

2. For a simple pendulum, a graph is plotted between its length (L) and the square of its time period (T²). The graph is:
   a) A parabola opening upwards
   b) A straight line passing through the origin
   c) A hyperbola
   d) A circle

3. A second's pendulum is a simple pendulum whose time period is:
   a) 1 second
   b) 2 seconds
   c) π seconds
   d) Determined by its length only

4. If the length of a simple pendulum is doubled, its time period will:
   a) Be halved
   b) Be doubled
   c) Increase by a factor of √2
   d) Decrease by a factor of √2

5. The effective length of a simple pendulum is the distance between:
   a) Point of suspension and the bottom of the bob
   b) Point of suspension and the top of the bob
   c) Point of suspension and the centre of gravity of the bob
   d) The ends of the string

6. To determine the time period of a pendulum accurately, the time should be measured for:
   a) One oscillation only
   b) Half an oscillation
   c) A large number of oscillations (e.g., 20 or 30)
   d) Any number of oscillations

7. If a simple pendulum experiment is performed inside a lift accelerating upwards, the time period will:
   a) Increase
   b) Decrease
   c) Remain unchanged
   d) Become infinite

8. The formula T = 2π√(L/g) for a simple pendulum is valid only if:
   a) The mass of the bob is large
   b) The length of the pendulum is small
   c) The amplitude of oscillation is small
   d) The experiment is done in a vacuum

9. From the slope 'm' of the L vs T² graph (L on y-axis, T² on x-axis), the acceleration due to gravity 'g' is given by:
   a) g = 4π² / m
   b) g = m / 4π²
   c) g = 4π² * m
   d) g = m * (2π)²

10. Which of the following is NOT a precaution for the simple pendulum experiment?
    a) Use a rigid support.
    b) Ensure the amplitude is very large for better visibility.
    c) Avoid air currents.
    d) Measure time for at least 20 oscillations.

Answer Key for MCQs:

1. c) Length of the pendulum
2. b) A straight line passing through the origin (assuming T² on Y-axis, L on X-axis, or vice-versa)
3. b) 2 seconds
4. c) Increase by a factor of √2 (Since T ∝ √L)
5. c) Point of suspension and the centre of gravity of the bob
6. c) A large number of oscillations (e.g., 20 or 30)
7. b) Decrease (Effective gravity g' = g+a, so T = 2π√(L/(g+a)) decreases)
8. c) The amplitude of oscillation is small
9. c) g = 4π² * m (Slope m = ΔL / ΔT². Since T² = (4π²/g)L, L/T² = g/4π², so m = g/4π²)
10. b) Ensure the amplitude is very large for better visibility. (Amplitude should be small)

Study these notes carefully. Understanding the theory, the formula, the graphical relationship, and the precautions/errors is key to tackling questions from this section in your exams. Let me know if any part needs further clarification.