Question

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\) (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\). (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) b (C) \(\mathrm{c}\) (D) \(\mathrm{d}\) Statement \(-1:\) If the length of a simple pendulum is increased by \(3 \%\), then the periodic time changes by \(1.5 \%\). Statement \(-2:\) Periodic time of a simple pendulum is proportional to its length. (A) a (B) \(b\) (C) \(c\) (D) d

Short answer

(b)

Step-by-step solution

Evaluate Statement -1

We are given that the length of the pendulum is increased by 3%. Let's calculate the new length (L_new) and its impact on the periodic time (P_new). L_new = L + 0.03L = 1.03L Now let's calculate the new periodic time (P_new) using the formula P_new = 2π√(L_new/g): P_new = 2π√((1.03L)/g) Next, let's find the percentage change in the periodic time: Percentage change = ( (P_new - P) / P ) × 100 Replacing P_new and P with their expressions: Percentage change = ( ( 2π√((1.03L)/g) - 2π√(L/g) ) / (2π√(L/g)) ) × 100 After simplification: Percentage change ≈ 1.5% Statement -1 is true.

Evaluate Statement -2

The formula for the period of a simple pendulum is: P = 2π√(L/g) From this formula, we can see that the periodic time (P) is proportional to the square root of the length (L) of the pendulum. Hence, Statement -2 is only partially true, as it does not mention the square root relation but only implies a direct proportion.

Identify the correct option

Statement -1 is true. Statement -2 is partially true, but not the correct explanation for Statement -1 because it does not provide the correct relationship between the periodic time and the length of the pendulum. Therefore, the correct option is: (b) Statement -1 is true; statement -2 is true, but statement -2 is not the correct explanation of Statement -1.