Question

In a damped oscillation with damping constant \(b\). The time taken for amplitude of oscillation to drop to half what is its initial value? (A) \((\mathrm{b} / \mathrm{m}) \ln 2\) (B) (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 amplitude of a damped oscillation to drop to half its initial value is \((2 \mathrm{~m} / \mathrm{b}) \ln 2\).

Step-by-step solution

Write down the decay constant formula for damped oscillations

The decay constant for damped oscillations can be written as: \(λ = \cfrac{b}{2m}\) where \(b\) is the damping constant and \(m\) is the mass of the object.

Find the time taken for amplitude to drop to half of its initial value

Considering the definition of half-life, the amplitude will reduce to half when the time elapsed is equal to the half-life \(t_{1/2}\). The general formula for half-life is: \(t_{1/2} = \cfrac{\ln 2}{λ}\)

Substitute the decay constant into the half-life formula

Now we will substitute the decay constant \(λ\) from Step 1 into the half-life equation from Step 2. \(t_{1/2} = \cfrac{\ln 2}{\cfrac{b}{2m}}\)

Simplify the equation

We will now simplify the equation to find the final formula. \(t_{1/2} = \cfrac{2m}{b} \ln 2\)

Compare the derived formula with the given options

Now compare the derived formula to the given choices: (A) \((\mathrm{b} / \mathrm{m}) \ln 2\) (B) (\(\mathrm{b} / 2 \mathrm{~m}) \ln 2\) (C) \((\mathrm{m} / \mathrm{b}) \ln 2\) (D) \((2 \mathrm{~m} / \mathrm{b}) \ln 2\) The derived formula matches option (D). Therefore, the time taken for the oscillation's amplitude to drop to half its initial value is \((2 \mathrm{~m} / \mathrm{b}) \ln 2\).

Key concepts

Damping Constant

In the study of oscillations, especially those that are damped, the **damping constant** plays a crucial role. It is denoted by the symbol \(b\) and acts to oppose the motion of the oscillating system. The damping effect is due to factors like friction or air resistance. The constant \(b\) essentially quantifies how resistive the medium is to the motion of the oscillator.

• A higher damping constant indicates that the oscillation will lose energy more quickly, causing the system to come to rest sooner.
• A smaller damping constant means that the oscillation will persist longer before stopping.

The damping constant can be appreciated in the context of the decay constant \(λ\), which relates the damping constancy to mass \(m\) of the system as follows:\[λ = \frac{b}{2m}\] Here the damping constant \(b\) is inversely proportional to the mass, suggesting that for heavier objects, the same damping force will have a lesser effect on halting the oscillation.

Half-Life

In the realm of damped oscillations, the concept of **half-life** emerges as a valuable metric. The half-life \(t_{1/2}\) measures the time taken for the amplitude of an oscillation to reduce to half its original value. This is synonymous with the idea of half-life found in other fields, such as radioactive decay.
To calculate the half-life for damped oscillations, we use the formula:\[t_{1/2} = \frac{\ln 2}{λ}\]with \(λ\) being the decay constant derived from the damping constant using \(λ = \frac{b}{2m}\).

• The calculation shows us how quickly the system experiences a significant reduction in motion.
• This understanding helps in designing oscillating systems with desired endurance or longevity.

In practice, using the above equation, we see that increasing the damping constant will decrease the half-life, meaning the system loses its energy at a quicker pace.

Amplitude Decay in Physics

Amplitude decay is a fundamental concept in physics that describes the reduction in the amplitude of an oscillating system over time. In damped oscillations, this decay is due to the presence of damping forces which consume the energy of the motion.
Key factors contributing to the amplitude decay include:

• The damping constant \(b\), which shows how strongly the damping forces act.
• The mass \(m\), where lighter objects will typically experience more significant amplitude reduction under the same damping force.

Mathematically, the relationship between amplitude \(A(t)\) and time \(t\) in a damped system can be expressed using an exponential decay function:\[A(t) = A_0 e^{-\frac{bt}{2m}}\]where:

• \(A_0\) is the initial amplitude.
• \(b\) is the damping constant,
• \(m\) is the mass of the oscillating object.

Through this relationship, we see how exponential functions capture the essence of amplitude reduction, hence describing how oscillations gradually diminish over time.