A pendulum takes 60 seconds to complete 24 oscillations. If its length is increased and it now takes 72 seconds for the same number of oscillations, what has happened to its time period?
It has increased to 3.0 seconds
To find out what has happened to the time period of the pendulum when its length is increased, let's go through the steps:
- First, calculate the initial time period of the pendulum. The time period \(T\) is given by the total time divided by the number of oscillations. Initially, the pendulum takes 60 seconds for 24 oscillations. Thus, the initial time period is: \(T_1 = \frac{60}{24} = 2.5 \text{ seconds}\).
- Now, let's calculate the new time period after the length is increased. The pendulum now takes 72 seconds for the same number of oscillations (24). Thus, the new time period is: \(T_2 = \frac{72}{24} = 3.0 \text{ seconds}\).
- Comparing both time periods, we can see that the time period has increased from 2.5 seconds to 3.0 seconds.
- The increase in time period could be due to an increase in the pendulum's length because the time period of a simple pendulum is given by the formula: \(T = 2 \pi \sqrt{\frac{L}{g}}\), where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity. This formula shows that the time period \(T\) is directly proportional to the square root of its length (\(L\)).
Therefore, the correct answer is that the time period has increased to 3.0 seconds.
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