A 65kg gymnast jumps on the trampoline. At the lowest point of her motion, the trampoline is depressed .50m below equilibrium. what is her approximate upward acceleration at this point?
To determine the approximate upward acceleration of the gymnast at the lowest point of her motion, we can make use of the concept of potential energy and Hooke's Law.
1. First, let's calculate the potential energy of the gymnast at the lowest point. The potential energy can be determined using the equation:
Potential energy (PE) = mass (m) × gravity (g) × height (h)
Given:
Mass (m) = 65kg (as mentioned in the question)
Height (h) = Depression of the trampoline = 0.50m
Potential energy (PE) = 65kg × 9.8m/s² × 0.50m
2. Now we need to find the approximate force exerted by the trampoline. According to Hooke's Law, the force is directly proportional to the displacement from equilibrium. The formula for the force exerted by the trampoline can be written as:
Force (F) = spring constant (k) × displacement (x)
Since it's at the lowest point, the displacement (x) would be equal to the depression of the trampoline, which is given as 0.50m.
3. The force exerted by the trampoline is the same as the weight of the gymnast because they are in equilibrium when the trampoline is fully depressed. Therefore, we can write:
Force (F) = weight (W) = mass (m) × gravity (g)
Substituting in the given mass:
Force (F) = 65kg × 9.8m/s²
4. Since the force exerted by the trampoline is equal to the spring force (F = kx), we can equate the two equations:
kx = 65kg × 9.8m/s²
Solving for the spring constant (k):
k = (65kg × 9.8m/s²) / (0.50m)
5. Finally, to find the approximate upward acceleration of the gymnast, we can use Newton’s second law:
Force (F) = mass (m) × acceleration (a)
Rearranging the equation and solving for acceleration (a):
a = F / m
Substituting in the known force and mass:
a = (65kg × 9.8m/s²) / 65kg
By following these steps, you can easily find the approximate upward acceleration of the gymnast at the lowest point of her motion.