A simple pendulum is made of a steel rod. If the temperature of the rod increases by 100 degrees Celsius, what is the percentage change in the time period of the pendulum?

Step 1: Calculate the change in temperature. The initial temperature is T_1 = 25^ C and the final temperature is T_2 = 125^ C. The change in temperature T is: T = T_2 - T_1 = 125^ C - 25^ C = 100^ C Step 2: Recall the formula for the time period of a simple pendulum and its dependence on length. The time period T of a simple pendulum is given by: T = 2 sqrt((L)/(g)) where L is the length of the pendulum and g is the acceleration due to gravity. Step 3: Relate the change in length to the change in temperature. When the temperature changes, the length of the steel rod changes due to thermal expansion. The new length L' is given by: L' = L(1 + T) where is the coefficient of linear expansion for steel. Step 4: Determine the new time period and the percentage change. The new time period T' will be: T' = 2 sqrt((L')/(g)) = 2 sqrt((L(1 + T))/(g)) = T sqrt(1 + T) The fractional change in time period is T_periodT = (T' - T)/(T) = sqrt(1 + T) - 1. Since T is very small, we can use the binomial approximation (1+x)^n ≈ 1+nx for small x. Here, x = T and n = (1)/(2). sqrt(1 + T) ≈ 1 + (1)/(2) T So, the fractional change in time period is approximately: T_periodT ≈ (1 + (1)/(2) T) - 1 = (1)/(2) T The percentage change in time period is: Percentage change = (1)/(2) T × 100\% Step 5: Substitute the given values and calculate the percentage change. Given _steel = 1.2 × 10^-5 /^ C and T = 100^ C. Percentage change = (1)/(2) × (1.2 × 10^-5 /^ C) × (100^ C) × 100\% Percentage change = 0.6 × 10^-5 × 100 × 100\% Percentage change = 0.6 × 10^-5 × 10^4 \% Percentage change = 0.6 × 10^-1 \% Percentage change = 0.06 \% Since the temperature increases, the length of the pendulum increases, which leads to an increase in the time period. Therefore, the change is positive. The final answer is 0.06%. Send me the next one 📸