Real world trampolines lose energy since they are damped springs with much internal friction. How much energy does the sumo wrestler lose on each bounce in this situation?
Information I was given:
mass=400 kg
gravity=10N/kg
k = 12000 N/m
I also know from the previous question each push of the legs drives the sumo wrestler 10 cm higher in the air, and on the first push, the trampoline went from 8 cm below its normal surface plane.
I know how to get the initial energy through mgh, but what about the final energy?
To calculate the energy loss on each bounce, you need to determine the initial and final energy of the sumo wrestler.
1. Initial energy:
The initial energy can be calculated using the equation: E_initial = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the height.
Given:
mass (m) = 400 kg
gravity (g) = 10 N/kg
initial height (h) = 8 cm = 0.08 m
E_initial = 400 kg * 10 N/kg * 0.08 m
E_initial = 320 N.m or Joules
2. Final energy:
To calculate the final energy, you need to take into account the change in height when the sumo wrestler bounces on the trampoline. Since each push raises the sumo wrestler 10 cm higher and the trampoline was initially 8 cm below its normal surface, the change in height would be 10 cm + 8 cm = 18 cm = 0.18 m.
The final energy can be calculated using the equation: E_final = m * g * h', where h' is the new height after the bounce.
Given:
mass (m) = 400 kg
gravity (g) = 10 N/kg
height change (h') = 0.18 m
E_final = 400 kg * 10 N/kg * 0.18 m
E_final = 720 N.m or Joules
To find the energy loss on each bounce, you can subtract the final energy from the initial energy:
Energy loss = E_initial - E_final
Energy loss = 320 N.m - 720 N.m
Energy loss = -400 N.m or Joules
Therefore, the sumo wrestler loses 400 N.m or 400 Joules of energy on each bounce.
To calculate the energy transferred to the trampoline on each bounce, we need to find the difference between the initial and final energy of the sumo wrestler.
First, let's calculate the initial energy. As you mentioned, the sumo wrestler is driven 10 cm higher in the air, so the height difference (h) is 10 cm or 0.1 m. The formula for gravitational potential energy (PE) is given by PE = m * g * h, where m is the mass, g is the acceleration due to gravity (10 N/kg), and h is the height difference. Plugging in the values:
PE_initial = m * g * h
= 400 kg * 10 N/kg * 0.1 m
= 400 J (joules)
Now let's find the final energy. Since real-world trampolines lose energy due to damping and internal friction, the final energy will be less than the initial energy. We can assume that the trampoline does not return to its original height after each bounce.
To determine the final energy, we need to know the position of the sumo wrestler after each bounce. Unfortunately, the information you provided doesn't specify the height difference or the number of bounces. Without this information, we cannot accurately determine the exact amount of energy lost on each bounce.
However, we can still provide a general approach for finding the final energy. After each bounce, the sumo wrestler will gain less and less height due to the energy losses on each bounce. You can assume that the height difference decreases with each subsequent bounce.
By considering the first bounce, since the trampoline was initially 8 cm below its normal surface plane and the sumo wrestler was pushed 10 cm higher, we can assume that the total upward displacement of the wrestler was (10 cm + 8 cm) = 18 cm or 0.18 m.
If we consider all subsequent bounces to have the same 0.18 m upward displacement, we can use the same formula as before to calculate the final energy:
PE_final = m * g * h
= 400 kg * 10 N/kg * 0.18 m
= 720 J (joules)
However, it's important to note that this value is an estimation, and the actual energy loss will depend on various factors such as the trampoline's damping and the sumo wrestler's motion.