In a damped oscillator the amplitude of vibrations of mass \(m=150\) grams falls by \(\frac{1}{e}\) 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 grams \(/ \mathrm{s}\) \((1) 6 \mathrm{~s}, \frac{e^{2}-1}{e^{2}} \times 100\) (2) 3s, \(\frac{e^{2}-1}{e^{2}} \times 100\) (3) 6s, \(\frac{e-1}{e} \times 100\) (4) 3s, \(\frac{e-1}{e} \times 100\)

The damping constant is a crucial factor in understanding damped oscillators. It represents the strength of the damping force, which opposes the motion of the oscillator. This force is typically due to viscous friction, such as air resistance or internal friction within the materials of the system. The damping constant is denoted by the symbol \( b \) and has units of mass per time (e.g., grams per second or kilograms per second).

In our exercise, the damping constant is given as 50 grams per second. However, we need it in kilograms per second for our calculations, so we convert it: \( 0.05 \) kg/s.

The damping constant is part of the exponential term in the amplitude decay equation for a damped oscillator:

\ A(t) = A_0 e^{-\frac{b}{2m}t} \

Here, \( b \) influences how quickly the amplitude of the oscillations decreases over time. A larger \( b \) leads to faster energy loss and greater reduction in the amplitude of the oscillations.

Mechanical energy in an oscillating system represents the sum of its kinetic and potential energies. In a damped oscillator, part of this energy is gradually lost due to the damping force. This energy loss manifests as a steady reduction in the amplitude of oscillations.

The mechanical energy of a damped oscillator decreases with the square of its amplitude:

\ E(t) = E_0 \left( \frac{A(t)}{A_0}\right)^2 \

Given the exponential decay of the amplitude in our example:

\ A(t) = A_0 e^{-\frac{b}{2m}t} \

We substitute this into the energy equation to determine the percentage mechanical energy loss over the specific time \( t_0 \):

\ \left( 1 - \left( \frac{A(t_0)}{A_0}\right)^2 \right) \times 100 \

Substituting \( A(t_0) = \frac{A_0}{e} \), we get:

\ \left( 1 - \left(\frac{1}{e}\right)^2 \right) \times 100 = \left( 1 - \frac{1}{e^2} \right) \times 100 = \frac{e^2 - 1}{e^2} \times 100 \

This result helps us understand how energy dissipates in the system due to damping.

Oscillations refer to any periodic motion that repeats itself in a regular cycle. In mechanical systems, oscillations can be found in pendulums, springs, and many other contexts. For an ideal oscillator with no damping, such as a weight on a perfect spring in a vacuum, the energy remains constant.

However, in practical scenarios, damping usually exists and is represented as a resisting force proportional to the velocity of the oscillating object. This damping causes the amplitude of the oscillations to decrease over time, resulting in damped oscillations.

For a damped oscillator, the displacement as a function of time is given by:

\ x(t) = A_0 e^{-\frac{b}{2m}t} \cos(\omega_d t + \phi) \

In this case, the term \( e^{-\frac{b}{2m}t} \) controls the damping, where \( b \) is the damping constant, \( m \) is the mass, and \( \omega_d \) is the damped angular frequency.

The angular frequency of the damped oscillator is slightly less than the natural frequency due to the energy lost to damping:

\ \omega_d = \sqrt{ \omega_0^2 - \left( \frac{b}{2m} \right)^2 } \

This expression illustrates how the frequency of oscillations is influenced by the damping constant and the mass. These concepts are essential in understanding the behavior of damped oscillators.