Answer :
The new period of the system, when the performer stands up, is approximately 2.487 seconds.
To solve this problem, we can use the formula for the period of a simple pendulum:
T = 2π√(L/g)
where:
T is the period of the pendulum,
L is the length of the pendulum, and
g is the acceleration due to gravity.
Initially, the performer is seated on the trapeze, and we need to find the initial length of the pendulum. Let's denote it as L_initial.
Given:
Period of the system when seated (T_initial) = 9.29 s
Rearranging the formula, we can solve for L_initial:
L_initial = (T_initial/2π)^2 * g
Now, let's calculate the new length of the pendulum when the performer stands up. We can use the concept of the center of mass to determine this.
Given:
Change in height (Δh) = 38.7 cm = 0.387 m
The new length of the pendulum (L_new) will be the sum of the initial length (L_initial) and the change in height (Δh).
L_new = L_initial + Δh
Finally, we can calculate the new period (T_new) using the formula for the period of a simple pendulum with the new length:
T_new = 2π√(L_new/g)
Let's plug in the given values and calculate the new period:
Using g = 9.8 m/s² (acceleration due to gravity), we have:
L_initial = (9.29/2π)² * 9.8 = 1.152 m
L_new = 1.152 + 0.387 = 1.539 m
T_new = 2π√(1.539/9.8) ≈ 2π√0.157 ≈ 2π * 0.396 ≈ 2.487 s
Therefore, the new period of the system, when the performer stands up, is approximately 2.487 seconds.
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