Question

In a damped oscillator the amplitude of vibrations of mass \(\mathrm{m}=150\) grams falls by \(\frac{1}{\mathrm{e}}\) times of its initial value in time \(t_{0}\) due to viscous forces. The time \(t_{0}\) and the percentage loss in mechanical energy during the above time interval \(t_{0}\) respectively are (Let damping constant be \(50 \mathrm{grams} / \mathrm{s}\) ) (1) 6s, \(\frac{\mathrm{e}^{2}-1}{\mathrm{e}^{2}} \times 100\) (2) 3s, \(\frac{e^{2}-1}{e^{2}} \times 100\) (3) 6s, \(\frac{e-1}{e} \times 100\) (4) 3s, \(\frac{e-1}{e} \times 100\)

Short answer

6 seconds, \( \frac{e^2 - 1}{e^2} \times 100 \).

Step-by-step solution

Determine the relationship between amplitude and time

The amplitude of a damped oscillator decreases over time according to the formula: \[ A(t) = A_0 e^{-b t / 2m} \] where \( A_0 \) is the initial amplitude, \( b \) is the damping constant, and \( m \) is the mass.

Use given information to find \( t_0 \)

It is given that the amplitude falls to \( \frac{1}{e} \) times its initial value in time \( t_0 \). Therefore, \[ e^{-b t_0 / 2m} = \frac{1}{e} \] Taking the natural logarithm on both sides: \[ -\frac{b t_0}{2m} = -1 \] \[ t_0 = \frac{2m}{b} \] Substituting the given values \( m = 150 \) grams \( = 0.15 \) kg and \( b = 50 \) grams/s \( = 0.05 \) kg/s: \[ t_0 = \frac{2 \times 0.15}{0.05} = 6 \text{ seconds} \]

Calculate the percentage loss in mechanical energy

The mechanical energy of a damped oscillator is proportional to the square of its amplitude. Therefore, the ratio of energies at times \( t = t_0 \) and \( t = 0 \) is: \[ \frac{E(t_0)}{E(0)} = \left( \frac{A(t_0)}{A_0} \right)^2 = \left( \frac{1}{e} \right)^2 = \frac{1}{e^2} \] The percentage loss in mechanical energy is then: \[ 100 \times \left(1 - \frac{1}{e^2} \right) = 100 \times \frac{e^2 - 1}{e^2} \]

Match with the given options

From the calculations, we find that \( t_0 = 6 \) seconds and the energy loss percentage is \( \frac{e^2 - 1}{e^2} \times 100 \). Thus, the correct answer is: (1) 6s, \( \frac{e^2 - 1}{e^2} \times 100 \).

Key concepts

Mechanical Energy in a Damped Oscillator

In a damped oscillator, mechanical energy is continuously lost due to forces such as friction or air resistance. This loss occurs because the system converts some of its mechanical energy into heat or other forms of energy.

Mechanical energy in a damped oscillator is proportional to the square of the amplitude of the oscillations. The formula is: \[ E \propto A^2 \] where \( E \) is the mechanical energy and \( A \) is the amplitude. So, when the amplitude decreases, the mechanical energy of the system also decreases.

In the given problem, the amplitude decreases to \[ \frac{1}{e} \] times its initial value in time \( t_0 \). Using this, we can find how mechanical energy changes over time by calculating the square of the amplitude ratio. When we know the decrease rate of the amplitude, we can determine the mechanical energy loss over the corresponding time period.

Damping Constant

The damping constant, often represented as \( b \), measures how quickly the oscillations in a damped system decrease over time. It's a constant related to forces that resist the motion of the oscillator, such as friction or air resistance.

The formula that describes the amplitude decay over time in the presence of damping is: \[ A(t) = A_0 e^{-bt/2m} \] Here, \( A_0 \) is the initial amplitude, \( b \) is the damping constant, \( t \) is the time, and \( m \) is the mass of the oscillating object. This equation shows that the amplitude decreases exponentially over time due to the damping forces.

In our problem, we are given a damping constant \( b = 50 \ \text{grams/second} = 0.05 \ \text{kg/second} \). Using the values provided, we determined the time \( t_0 \) in which the amplitude drops to \( \frac{1}{e} \) times its initial value through the relationship \[ t_0 = \frac{2m}{b} = 6 \ \text{seconds} \]. This helps us understand the rate at which the system loses its energy.

Amplitude Decay

Amplitude decay describes how the oscillations' maximum extent from equilibrium decreases over time in a damped oscillator. This decrease happens due to the damping forces that slow down the motion of the oscillating mass.

The decay of amplitude can be quantified through the exponential decay formula: \[ A(t) = A_0 e^{-bt/2m} \] In this formula, \( A_0 \) is the initial amplitude, \( A(t) \) is the amplitude at time \( t \), \( b \) is the damping constant, and \( m \) is the mass. It shows that the amplitude decreases exponentially as time passes.

For our specific problem, we know the amplitude falls to \( \frac{1}{e} \) times its initial value in 6 seconds. This exponential relationship is crucial in calculating other properties of the system, such as the mechanical energy loss over time. Understanding how the amplitude decays helps us predict the behavior of the oscillator more effectively.