Question

In a damped oscillation with damping constant \(b\). The time taken for its mechanical energy to drop to half. What is its value? (A) \((\mathrm{b} / \mathrm{m}) \ln 2\) (B) \((\mathrm{b} / 2 \mathrm{~m}) \ln 2\) (C) \((\mathrm{m} / \mathrm{b}) \ln 2\) (D) \((2 \mathrm{~m} / \mathrm{b}) \ln 2\)

Short answer

The time taken for the mechanical energy of a damped oscillation to drop to half is (C) \((\mathrm{m} / \mathrm{b}) \ln 2\).

Step-by-step solution

Understand damped oscillation and its relation to mechanical energy

In a damped oscillation, the mechanical energy of the system gradually dissipates due to various reasons, such as friction or air resistance. As the energy decreases, the amplitude of oscillation also decreases. The damping constant 'b' defines how quickly the energy of the system is dissipated. The mechanical energy of the damped oscillator is given by: \(E(t) = E_0 e^{- \frac{b}{m} t}\) where E(t) is the mechanical energy at time 't', E_0 is the initial mechanical energy, 'b' is the damping constant, 'm' is the mass of the oscillator, and 't' is the time.

Set up the inequality for half mechanical energy

We have to find the time 't' when the mechanical energy of the oscillator drops to half of its initial value. We can set up the equation as follows: \(\frac{1}{2}E_0 = E_0 e^{- \frac{b}{m} t}\)

Solve for time 't'

We now solve the equation for the time 't': \(\frac{1}{2} = e^{- \frac{b}{m} t}\) Take the natural logarithm of both sides: \(\ln\frac{1}{2} = - \frac{b}{m} t\) Now, isolate t on one side: \(t = -\frac{m}{b} \ln\frac{1}{2}\) We know that \(\ln\frac{1}{2} = -\ln 2\): \(t = \frac{m}{b} \ln 2\) The answer is: (C) \((\mathrm{m} / \mathrm{b}) \ln 2\)

Key concepts

Mechanical Energy

Mechanical energy in any system consists of two primary forms: kinetic and potential energy. In the context of oscillations, mechanical energy can be visualized as the sum of the energy stored in motion and the energy stored in position. For example, in a spring-mass system, the potential energy is stored in the spring's compression or extension, while the kinetic energy comes from the mass moving at a certain velocity.

In a damped oscillation, the mechanical energy doesn't remain constant. Instead, it decreases over time due to effects like friction or air resistance. This loss of energy manifests as a reduction in the amplitude of the oscillation. The equation for mechanical energy in a damped system is:

• The energy at any time t is given by: \( E(t) = E_0 e^{- \frac{b}{m} t} \)
• Here, \( E_0 \) is the initial energy, and \( e^{- \frac{b}{m} t} \) shows how quickly energy decays depending on the damping constant \( b \) and mass \( m \).

Hence, understanding mechanical energy in a damped system requires appreciating how internal and external forces cause energy dissipation over time.

Damping Constant

The damping constant, often represented by 'b', plays a critical role in the process of damped oscillations. It is a measure of how quickly the system loses energy, which can visually be interpreted as how rapidly the oscillations come to a halt. The larger the damping constant, the quicker the oscillations cease.

In the formula for mechanical energy \( E(t) = E_0 e^{-\frac{b}{m} t} \), \( b \) directly influences the rate of exponential decay. This means that:

• A higher damping constant implies a rapid energy loss.
• A lower damping constant means the system will lose mechanical energy more slowly.

Choosing the appropriate damping constant is crucial in applications like automobile shock absorbers, where comfort is a priority, and in seismology, where controlling oscillations is important.

Exponential Decay

Exponential decay is a mathematical concept that represents the process by which quantities reduce at a rate proportional to their current value. In damped oscillations, mechanical energy decreases exponentially over time, symbolized by the function \( e^{-\frac{b}{m} t} \).

This sort of decay implies that:

• The mechanical energy drops rapidly at first, then more slowly over time.
• Each decrease takes the system closer to a minimal energy state but never quite reaches zero.
• The decay constant \( -\frac{b}{m} \) dictates how quickly this reduction occurs.

Understanding exponential decay in the context of damped oscillations highlights why certain systems, like clocks and pendulums, eventually come to a stop unless acted upon by an external energy source.

Initial Energy

Initial energy, denoted by \( E_0 \), is the total mechanical energy a system possesses at time \( t = 0 \). In terms of oscillations, it represents the total potential and kinetic energy the system starts with before any energy loss occurs.

In a damped oscillation, knowing the initial energy is essential because:

• It serves as a reference point to gauge how much energy has been dissipated over time.
• The rate at which this initial energy diminishes provides insight into the effectiveness of damping.

Consider the equation \( E(t) = E_0 e^{-\frac{b}{m} t} \). Here, the role of initial energy is evident. As \( t \) increases, \( e^{-\frac{b}{m} t} \) approaches zero, causing \( E(t) \) to become much smaller than \( E_0 \), illustrating the concept of energy dissipation. Knowing the initial energy allows for implementing modifications, like altering the damping constant, to achieve the desired behavior of the system.