Question

A rigid body rotates about a fixed axis with variable angular velocity $(\alpha - \beta t)$ at time $t$, where $\alpha$ and $\beta$ are constants. The angle through which it rotates before it comes to rest is ______.

• A. $\frac{\alpha}{\beta}$
• B. $\frac{\alpha^2}{\beta}$
• C. $\frac{\alpha^2}{2\beta}$
• D. $\frac{\alpha}{2\beta}$

Hint:

This is functionally identical to the 1D kinematic equation $v^2 = u^2 + 2aS$.
Here, initial velocity $\omega_0 = \alpha$, final velocity $\omega_f = 0$, and angular acceleration is the derivative of $\omega$, which is $-\beta$.
$0^2 = \alpha^2 + 2(-\beta)\theta \implies 2\beta\theta = \alpha^2 \implies \theta = \frac{\alpha^2}{2\beta}$. Much faster!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the angular velocity ($\omega$) as a function of time. We must find the total angular displacement ($\theta$) from $t=0$ until the exact moment the body stops rotating.

Step 2: Detailed Explanation:
The angular velocity is given by:
$\omega = \alpha - \beta t$
1. Find the time when the body comes to rest:
The body comes to rest when its angular velocity drops to zero ($\omega = 0$).
$0 = \alpha - \beta t$
$\beta t = \alpha \implies t = \frac{\alpha}{\beta}$
So, the body rotates from $t = 0$ to $t = \frac{\alpha}{\beta}$.
2. Calculate the angular displacement ($\theta$):
Angular displacement is the definite integral of angular velocity with respect to time:
$\theta = \int_{0}^{t} \omega \, dt$
$\theta = \int_{0}^{\alpha/\beta} (\alpha - \beta t) \, dt$
Integrate the function term by term:
$\theta = \left[ \alpha t - \frac{\beta t^2}{2} \right]_0^{\alpha/\beta}$
Evaluate the integral at the upper limit (the lower limit evaluates to 0):
$\theta = \alpha \left( \frac{\alpha}{\beta} \right) - \frac{\beta}{2} \left( \frac{\alpha}{\beta} \right)^2$
$\theta = \frac{\alpha^2}{\beta} - \frac{\beta}{2} \left( \frac{\alpha^2}{\beta^2} \right)$
$\theta = \frac{\alpha^2}{\beta} - \frac{\alpha^2}{2\beta}$
Find a common denominator to subtract:
$\theta = \frac{2\alpha^2}{2\beta} - \frac{\alpha^2}{2\beta} = \frac{\alpha^2}{2\beta}$

Step 3: Final Answer:
The angle rotated is $\frac{\alpha^2}{2\beta}$, matching option (c).