Two sirnpie pendulums of lengths in the ratio 9: 1 start oscillating at t= 0, They will again be In phase after the shorter pendulum has made n oscillations. The value of n is (1)2 (2) 1/2 (3)3/2 (4)3

To determine how many oscillations the shorter pendulum makes before both pendulums are in phase again, we need to analyze the relationship between their lengths and the time period of their oscillations. The lengths of the pendulums are in the ratio of 9:1, which means if we denote the length of the shorter pendulum as L, the longer pendulum will have a length of 9L.

Understanding the Time Period of a Pendulum

The time period (T) of a simple pendulum is given by the formula:

T = 2π√(L/g)

where L is the length of the pendulum and g is the acceleration due to gravity. Since both pendulums are in the same gravitational field, we can focus on the lengths to find their respective time periods.

Calculating the Time Periods

• The time period of the shorter pendulum (T1) is:

T1 = 2π√(L/g)

• The time period of the longer pendulum (T2) is:

T2 = 2π√(9L/g) = 2π√(9)√(L/g) = 6π√(L/g)

Finding the Ratio of Time Periods

Now, let's find the ratio of the time periods:

Ratio = T2/T1 = (6π√(L/g)) / (2π√(L/g)) = 3

This means that the longer pendulum takes 3 times longer to complete one oscillation compared to the shorter pendulum.

Determining When They Are In Phase Again

For the two pendulums to be in phase again, they must complete a whole number of oscillations. Let’s denote the number of oscillations made by the shorter pendulum as n. In the time it takes for the shorter pendulum to complete n oscillations, the longer pendulum will complete:

n/3 oscillations.

To find when they are both in phase, n must be a multiple of 3, since the longer pendulum completes one oscillation for every 3 of the shorter pendulum. The smallest whole number that satisfies this condition is 3.

Final Answer

Thus, the value of n, which represents the number of oscillations made by the shorter pendulum before both pendulums are in phase again, is 3.