Question

In a simple pendulum experiment for the determination of acceleration due to gravity, the error in the measurement of the length of the pendulum is 1% and the error in the measurement of the time period is 2%. The error in the estimation of acceleration due to gravity is: 

• A. \( 1\% \)
• B. \( 3\% \)
• C. \( 4\% \)
• D. \( 5\% \)

Hint:

For error propagation in division and power functions, use the formula: \[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \frac{\Delta T}{T} \] where the exponent contributes as a multiplying factor to the percentage error.

Answer 1

The Correct Option is D

Step 1: Understanding the Relationship The acceleration due to gravity \( g \) in a simple pendulum is given by: \[ g = \frac{4\pi^2 L}{T^2} \] Taking the percentage error on both sides: \[ \frac{\Delta g}{g} = \frac{\Delta L}{L} + 2 \frac{\Delta T}{T} \]

Step 2: Substituting Given Values Given: \[ \frac{\Delta L}{L} = 1\%, \quad \frac{\Delta T}{T} = 2\% \] \[ \frac{\Delta g}{g} = 1\% + 2(2\%) = 1\% + 4\% = 5\% \] Thus, the error in the estimation of acceleration due to gravity is: \[ \mathbf{5\%} \]