Equilibrium and Trampoline Stretch Calculation
How can we calculate the distance the trampoline stretches when the gymnast stands on it at rest?
When a gymnast jumps on a trampoline, her feet reach a maximum height of 3.48 m above the trampoline and stretch it 70.0 cm down. How far does the trampoline stretch when she stands on it at rest?
Calculation Method
To determine the distance the trampoline stretches when the gymnast stands on it at rest, we can use Hooke's law, which describes the behavior of a spring-like object when stretched or compressed. The force applied to the trampoline is balanced by the force of the trampoline pushing back, resulting in equilibrium.
Let's break down the process step by step:
- Determine the spring constant (k) using the gymnast's maximum height reached above the trampoline.
- Calculate the displacement (x_rest) when the gymnast is at rest based on the calculated spring constant.
Step 1: Calculate the Spring Constant (k)
First, we need to calculate the spring constant (k) using the given information that the gymnast's feet stretch the trampoline 70.0 cm (0.7 m) down and she reaches a maximum height of 3.48 m above the trampoline.
Since the trampoline follows Hooke's law, we can express the force exerted on the trampoline at maximum displacement as: F_max = m * g. This force is equal to the weight of the gymnast.
Using the potential energy gained by the gymnast at the maximum height, we can solve for k: k = (2 * m * g * h) / x_max². Substitute the known values to find the spring constant, k.
Step 2: Calculate Displacement at Equilibrium (x_rest)
At equilibrium, the force applied to the trampoline is balanced by the force of the trampoline pushing back. This results in equilibrium, where F_rest = m * g.
Using Hooke's law equation (F_rest = k * x_rest), rearrange the formula to solve for x_rest: x_rest = F_rest / k. Substitute the calculated spring constant and known values to find the displacement when the gymnast stands on the trampoline at rest.