Kater's Pendulum Experiment

Determination of Acceleration Due to Gravity Using Kater's Reversible Pendulum

1. Aim

To determine the acceleration due to gravity (g) at a place using Kater's reversible pendulum and to verify that the time period is the same when suspended from either knife edge.

2. Apparatus Used

  • Kater's reversible pendulum (metal bar with two knife edges and adjustable masses)
  • Precision stopwatch (least count 0.01s)
  • Meter scale (least count 1mm)
  • Vernier calipers
  • Spirit level
  • Knife edge support stand
  • Screwdriver for mass adjustment

3. Diagram

Figure: Kater's reversible pendulum showing two knife edges (K1 and K2) and adjustable masses

4. Theory

Kater's reversible pendulum is a compound pendulum that can be suspended from either of two knife edges. When properly adjusted, it has equal time periods about both suspension points, eliminating the need to locate the center of gravity precisely.

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

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

Where:

  • \( T \) = Time period of oscillation
  • \( h \) = Distance from pivot to center of gravity
  • \( k \) = Radius of gyration about center of gravity
  • \( g \) = Acceleration due to gravity

For Kater's reversible pendulum with two knife edges at distances h1 and h2 from the CG:

\[ T_1 = 2\pi \sqrt{\frac{k^2 + h_1^2}{gh_1}} \] \[ T_2 = 2\pi \sqrt{\frac{k^2 + h_2^2}{gh_2}} \]

When \( T_1 = T_2 = T \), For equal time periods, this simplifies to:

\[ g = \frac{8\pi^2(h_1 + h_2)}{T^2} \]

Special Note: The unique feature of Kater's pendulum is that when the time periods about both knife edges are made equal, the distance between the knife edges (h1 + h2) becomes equal to the length of an equivalent simple pendulum.

5. Formula

The acceleration due to gravity is calculated using:

\[ g = \frac{8\pi^2(h_1 + h_2)}{T^2} \]

Where:

  • \( h_1 \) = Distance from knife edge K1 to CG
  • \( h_2 \) = Distance from knife edge K2 to CG
  • \( T \) = Common time period when \( T_1 = T_2 \)
  • \( h_1 + h_2 \) = Distance between the two knife edges

6. Procedure

  1. Set up the kater's reversible pendulum on the knife edge stand ensuring it is horizontal (use spirit level).
  2. Suspend the pendulum from knife edge K1 and measure time for 20 oscillations (T1).
  3. Repeat for knife edge K2 to measure T2.
  4. Adjust the movable weights until T1 ≈ T2 (difference ≤ 0.1s).
  5. When satisfied with adjustment, make fine measurements of T1 and T2 (time for 50 oscillations).
  6. Measure the distance between the two knife edges (h1 + h2).
  7. Determine the position of center of gravity by balancing the pendulum horizontally.
  8. Measure h1 and h2 from CG to each knife edge.
  9. Record all observations in the tabular form.
  10. Calculate g using the formula.

Adjustment Procedure: The movable weights are adjusted until the time periods about both knife edges are equal. This may require several iterations of:

  1. Measuring T1 and T2
  2. Moving the adjustable mass slightly
  3. Re-measuring the time periods

7. Observation Table

S.No. Knife Edge Time for 50 oscillations (s) Mean Time Period (T) (s) Distance from CG (h) (cm)
Trial 1 Trial 2 Trial 3
1 K1 h1 =
2 K2 h2 =

Distance between knife edges (h1 + h2) = _____ cm = _____ m

8. Calculations

  1. Calculate mean time period for each knife edge:
    \[ T = \frac{\text{Mean time for 50 oscillations}}{50} \]
  2. Ensure \( T_1 \) and \( T_2 \) are nearly equal (difference ≤ 0.5%).
  3. Calculate g using:
    \[ g = \frac{8\pi^2(h_1 + h_2)}{T^2} \]
    where T is the average of T1 and T2
  4. For more accurate results, use:
    \[ g = 8\pi^2 \left[ \frac{h_1 + h_2}{T_1^2 + T_2^2} + \frac{h_1 - h_2}{T_1^2 - T_2^2} \right]^{-1} \]
  5. Compare with standard value and calculate percentage error.

Optional Graph Plot

For understanding the relationship, plot:

  • T vs h: Shows how period changes with suspension point
  • T2h vs h2: Should give a straight line from which k (radius of gyration) can be determined

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 = _____ %
  • The time periods T1 and T2 were found to be equal within experimental error, verifying the principle of Kater's pendulum

10. Precautions

  • Ensure the knife edges are sharp and properly aligned
  • Amplitude of oscillation should be small (≤ 2°)
  • Time a large number (50) of oscillations to reduce timing errors
  • Measure distances carefully from knife edges to CG
  • Avoid air currents that may affect oscillations
  • Make fine adjustments to movable masses for equalizing periods
  • Ensure pendulum swings in one plane without wobbling
  • Verify that the support is rigid and vibration-free

11. Viva Voce Questions

1. What is a reversible pendulum? Why is it called 'reversible'?
2. What is the advantage of Kater's reversible pendulum over a simple pendulum?
3. Derive the expression for time period of a compound pendulum.
4. Why do we adjust the pendulum to have equal time periods about both knife edges?
5. What is the significance of the radius of gyration in this experiment?
6. How does the position of the center of gravity affect the time period?
7. What would happen if the knife edges are not properly aligned?
8. How does this method eliminate the need for precise location of the center of gravity?
9. What is the relationship between Kater's reversible pendulum and an equivalent simple pendulum?
10. What are the practical applications of this experiment?
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