A body with mass 0.30 kg hangs by a spring with force constant 50.0 N/m. By what
factor is the frequency of oscillation reduced if the oscillation is damped and reaches
1/e of its original amplitude in 100 oscillations?
To find the factor by which the frequency of oscillation is reduced, we need to understand the concept of damping and its effect on the oscillation.
Damping occurs when there is a dissipative force, such as friction or air resistance, that opposes the motion of the body. This dissipative force reduces the amplitude of the oscillation over time.
The rate at which the amplitude decreases can be described by the decay of the amplitude over time. In this specific problem, we are given that the oscillation reaches 1/e (approximately 0.37) of its original amplitude in 100 oscillations.
The decay of the amplitude can be modeled using the equation:
A(t) = A₀ * e^(-γt)
Where:
A(t) is the amplitude of the oscillation at time t
A₀ is the initial amplitude (before any damping)
γ is the damping coefficient
We know that A(t) = A₀/e when t = 100.
Substituting these values into the equation, we have:
A₀/e = A₀ * e^(-γ * 100)
Dividing both sides by A₀:
1/e = e^(-γ * 100)
Taking the natural logarithm (ln) of both sides:
ln(1/e) = ln(e^(-γ * 100))
Simplifying:
-1 = -γ * 100
Dividing both sides by -100:
γ = 0.01
The damping coefficient (γ) is equal to 0.01.
The frequency of oscillation can be determined using the following formula:
f = 1 / (2π * √(m/K))
Where:
f is the frequency of oscillation
m is the mass of the body
K is the force constant of the spring
Substituting the given values into the formula:
f₀ = 1 / (2π * √(0.3 kg / 50 N/m))
f₀ = 0.827 Hz
The initial frequency of oscillation (f₀) is 0.827 Hz.
With damping, the new frequency (f) can be calculated using:
f = f₀ * √(1 - (γ / (2π))^2)
Substituting the values:
f = 0.827 Hz * √(1 - (0.01 / (2π))^2)
f ≈ 0.787 Hz
The new frequency of oscillation (f) is approximately 0.787 Hz.
To find the factor by which the frequency is reduced, we can calculate the ratio of the new frequency to the initial frequency:
factor = f / f₀
factor = 0.787 Hz / 0.827 Hz
factor ≈ 0.951
Therefore, the frequency of oscillation is reduced by a factor of approximately 0.951.