Problem 22 - Master Resource Book in JEE Main Physics

Question

The damping force of an oscillating particle is observed to be proportional to velocity. The constant of proportionality can be measured in (a) \(\mathrm{kg} \mathrm{s}^{-1}\) (b) \(\mathrm{kg} \mathrm{s}\) (c) \(\mathrm{kg} \mathrm{ms}^{-1}\) (d) \(\mathrm{kg} \mathrm{m}^{-1} \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

The constant of proportionality is measured in \( \mathrm{kg} \mathrm{s}^{-1} \).

Step 1: Understanding Proportionality

The damping force is proportional to velocity, meaning you can express it as \( F_d = b \cdot v \), where \( F_d \) is the damping force, \( v \) is the velocity, and \( b \) is the constant of proportionality.

Step 2: Identifying Dimensional Formula for Force and Velocity

The dimensional formula for force is \( \, [F] = [M][L][T]^{-2} \, \) (mass times length per square time). The dimensional formula for velocity is \( \, [v] = [L][T]^{-1} \, \) (length per time).

Step 3: Solving for the Constant of Proportionality

To find the dimensional formula for the constant \( b \), rearrange the proportionality equation in terms of \( b \): \( \, b = \frac{F_d}{v} \, \). Substitute the dimensional formulas: \[ \, [b] = \frac{[F]}{[v]} = \frac{[M][L][T]^{-2}}{[L][T]^{-1}} = [M][T]^{-1} \, \]

Step 4: Matching Units with Options

The dimensional formula \( \, [M][T]^{-1} \, \) corresponds to the units of \( \, \mathrm{kg} \mathrm{s}^{-1} \, \). Thus, the correct unit from the given options is (a) \( \mathrm{kg} \mathrm{s}^{-1} \).

Key Concepts

Proportionality Constant

A proportionality constant is a factor that relates two quantities that are proportional to each other. In the case of the damping force of an oscillating particle, this constant is denoted as \( b \). When a damping force \( F_d \) is proportional to velocity \( v \), it means you can express this relationship as \( F_d = b \cdot v \). Here, \( b \) is the constant of proportionality. Importance in Damping Force
The proportionality constant \( b \) plays a crucial role in understanding how the force interacts with the velocity. As the velocity of the particle changes, the magnitude of the damping force changes in direct proportion, thanks to the constant \( b \).- A larger \( b \) indicates a greater damping effect, meaning the force will significantly affect the particle at any given velocity.- A smaller \( b \) means that the velocity's influence on the damping force is less prominent.

Dimensional Analysis

Dimensional analysis is a powerful tool used to understand the relationships between physical quantities by examining their dimensions. It's about breaking down quantities into their basic units of measurement, such as mass \([M]\), length \([L]\), and time \([T]\). Applying to Forces and Velocity
In our exercise, we analyze the damping force and velocity to find the correct unit for the constant of proportionality. - The force has a dimensional formula of \([F] = [M][L][T]^{-2}\), indicating it involves mass, distance, and time squared.- Velocity has a dimension of \([v] = [L][T]^{-1}\), combining distance and time. Deriving the Constant's Dimension
To find the dimension of \( b \), you divide the force by velocity: \[[b] = \frac{[M][L][T]^{-2}}{[L][T]^{-1}} = [M][T]^{-1}\]This result means that the constant \( b \) is associated with mass per time, consistent with the unit \( \mathrm{kg} \mathrm{s}^{-1} \), confirming option (a) in our list.

Oscillating Particle

An oscillating particle moves back and forth in a regular periodic fashion. It’s a classic example in physics for studying motion and forces such as damping. Importance of Damping in Oscillations
Damping is a force that reduces the energy of the oscillating system, usually resulting in the gradual cessation of motion. It occurs due to various factors such as friction and resistance.- Damping forces are crucial in reducing amplitude over time, ensuring controlled and stable oscillations.- In a practical sense, systems with controlled damping, like car shock absorbers, enhance comfort and performance by managing oscillations effectively.Velocity's Role in Damping
For oscillating particles, the damping force is directly influenced by the velocity. As the particle's speed increases, the damping force also increases, if correlated through the proportionality constant \( b \). This relationship helps in predicting and managing how systems will behave under oscillating conditions.