Question

A 210-kg woorden raft floats on a lake. When a 75-kg man stands on the raft, it sinks 5.0 cm deeper into the water. When he steps off, the raft vibraties for a while. A. What is the frequency of vibration? B. What is the total energy of vibration? Hint: ignore damping.

Answer 1

Answer:

A: The frequency of the vibration is 1.3329 Hz

B: The total energy of the vibration is  18.39375 J

Explanation:

The force of the man his weight causes the raft to sink, and that causes the water to put a larger upward force on the raft.  This extra force is a restoring force, because it is in the opposite direction of the force put on the raft by the man.  Then when the man steps off, the restoring force pushes upward on the raft, and thus the raft – water system acts like a spring, with a spring constant found as follows:

k= F/x = ((75 kg) * (9.81 m/s²))/(5*10^-2 m) = 14715 N/m

The frequency of the vibration is determined by the spring constant (k) and the mass of the raft (210kg).

fn = 1/2π * √(k/m)  = 1/2π * √(14715 / 210) = 1.3329 Hz

The frequency of the vibration is 1.3329 Hz

b)

Since the gravitational potential energy can be ignored, the total energy will be :

Etot = 1/2 k* A² = 1/2 * (14715 )*(0.05)² = 18.39375 J

The total energy of the vibration is  18.39375 J