Question

A rigid body rotates about a fixed axis with variable angular velocity \( \omega = \alpha - \beta t \) at time \( t \), where \( \alpha, \beta \) are constants. The angle through which it rotates before it stops is:

• A. \( \frac{\alpha^2}{2\beta} \)
• B. \( \frac{\alpha^2 - \beta^2}{2\alpha} \)
• C. \( \frac{\alpha^2 - \beta^2}{2\beta} \)
• D. \( \frac{(\alpha - \beta)\alpha}{2} \)

Hint:

For angular motion, the equation: \[ \omega^2 = \omega_0^2 + 2\beta \theta \] is analogous to linear kinematics and helps in solving rotation problems efficiently.

Answer 1

The Correct Option is A

Step 1: {Determine stopping condition}
The angular velocity is given as: \[ \omega = \alpha - \beta t \] The body stops when \( \omega = 0 \), so: \[ 0 = \alpha - \beta t \] Solving for \( t \): \[ t = \frac{\alpha}{\beta} \] Step 2: {Use kinematic equation for angular motion}
The equation: \[ \omega^2 = \omega_0^2 + 2\beta \theta \] Substituting \( \omega_0 = \alpha \) and \( \omega = 0 \): \[ 0 = \alpha^2 - 2\beta \theta \] Step 3: {Solve for \( \theta \)}
\[ \theta = \frac{\alpha^2}{2\beta} \] Thus, the correct answer is (A) \( \frac{\alpha^2}{2\beta} \).

Answer 2

Given:
The angular velocity of a rigid body varies with time as:
\[ \omega(t) = \alpha - \beta t \] Where \( \alpha, \beta \) are positive constants.

Objective:
Find the total angular displacement (angle rotated) before the body comes to rest.

Step 1: Find the time when the body stops rotating
The body stops when angular velocity becomes zero:
\[ \omega = \alpha - \beta t = 0 \Rightarrow t = \frac{\alpha}{\beta} \]

Step 2: Use integration to find angular displacement
Angular displacement \( \theta \) is given by:
\[ \theta = \int_0^{t} \omega(t)\, dt = \int_0^{\alpha/\beta} (\alpha - \beta t)\, dt \]
Now integrate:
\[ \theta = \left[ \alpha t - \frac{\beta t^2}{2} \right]_0^{\alpha/\beta} = \alpha \cdot \frac{\alpha}{\beta} - \frac{\beta}{2} \cdot \left( \frac{\alpha^2}{\beta^2} \right) = \frac{\alpha^2}{\beta} - \frac{\alpha^2}{2\beta} = \frac{\alpha^2}{2\beta} \]

Final Answer:
The angle through which the body rotates before it stops is:
\[ \boxed{\frac{\alpha^2}{2\beta}} \]