A 220 kg wooden raft sits on a lake. When a 75 kg man stands on the raft, it sinks 4 cm deeper into the water. When he steps off, the raft vibrates for a while. a) What is the frequency of the raft? b) What is the total energy of the vibration?
To answer these questions, we need to understand the concepts of frequency and energy in the context of the vibrating raft.
a) To determine the frequency of the raft's vibrations, we can use the information given about the change in depth when the man stood on the raft. The change in depth of the raft can be related to the frequency using the equation:
Δy = (2π²f²/m) * Δx
Where:
Δy = Change in depth of the raft (4 cm)
f = Frequency of the vibration (what we're trying to find)
m = Mass of the raft + man (220 kg + 75 kg = 295 kg)
Δx = Displacement of depth caused by the man standing on the raft
We are given that the depth of the raft changes by 4 cm (Δy = 0.04 m) when the man stands on it. Let's assume that the displacement caused by the man is half of this change, so Δx = 0.02 m.
Plugging in these values into the equation, we have:
0.04 = (2π²f²/295) * 0.02
Now, we can solve for the frequency (f):
(2π²f²/295) = 0.04 / 0.02
2π²f² = 2
f² = 1 / π²
f = √(1 / π²)
f ≈ 0.564 Hz
Therefore, the frequency of the raft's vibrations is approximately 0.564 Hz.
b) To determine the total energy of the vibration, we need to consider the potential energy (PE) and the kinetic energy (KE) associated with the oscillations. The total energy (TE) is the sum of these two:
TE = PE + KE
The potential energy at the maximum displacement (amplitude) of the vibration is given by:
PE = 0.5 * k * (amplitude)²
Where:
k = Spring constant (we don't have this information)
(amplitude) = Maximum displacement caused by the vibrations (also not given)
To calculate the kinetic energy, we need to know the velocity of the mass when it is moving with its maximum displacement. However, this information is not provided in the question. Thus, we cannot determine the total energy of the vibration without additional details.