A rigid body rotates about a fixed axis with variable angular velocity equal to (a - bt) at time t where a and b are constants. The angle through which it rotates before it comes to rest is

Options

A. `a^2/b`
B. `a^2/(2b)`
C. `a^2/(4b)`
D. `a^2/(2b^2)`

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Understand the given angular velocity
The angular velocity of the rigid body is given by the equation:
\[ \omega(t) = a - bt \]
where \( a \) and \( b \) are constants, and \( t \) is the time.

### Step 2: Determine when the body comes to rest
The body comes to rest when the angular velocity \( \omega \) becomes zero. Therefore, we set:
\[ a - bt = 0 \]
Solving for \( t \):
\[ bt = a \implies t = \frac{a}{b} \]

### Step 3: Relate angular velocity to angular displacement
We know that angular velocity is related to angular displacement \( \theta \) by the equation:
\[ \omega = \frac{d\theta}{dt} \]
This can be rearranged to express \( d\theta \):
\[ d\theta = \omega \, dt \]

### Step 4: Substitute the expression for angular velocity
Substituting the expression for \( \omega \) into the equation gives:
\[ d\theta = (a - bt) \, dt \]

### Step 5: Integrate to find the total angle rotated
We will integrate \( d\theta \) from \( 0 \) to \( \theta \) and \( dt \) from \( 0 \) to \( t \):
\[ \theta = \int_0^t (a - bt) \, dt \]
Calculating the integral:
\[ \theta = \int_0^t a \, dt - \int_0^t bt \, dt \]
The first integral evaluates to:
\[ \int_0^t a \, dt = at \]
The second integral evaluates to:
\[ \int_0^t bt \, dt = b \frac{t^2}{2} \]
Thus, we have:
\[ \theta = at - \frac{b t^2}{2} \]

### Step 6: Substitute \( t = \frac{a}{b} \)
Now substitute \( t = \frac{a}{b} \) into the equation for \( \theta \):
\[ \theta = a\left(\frac{a}{b}\right) - \frac{b}{2}\left(\frac{a}{b}\right)^2 \]
Calculating each term:
\[ \theta = \frac{a^2}{b} - \frac{b}{2} \cdot \frac{a^2}{b^2} \]
\[ = \frac{a^2}{b} - \frac{a^2}{2b} \]
Combining the terms:
\[ \theta = \frac{2a^2}{2b} - \frac{a^2}{2b} = \frac{a^2}{2b} \]

### Final Result
Thus, the angle through which the rigid body rotates before it comes to rest is:
\[ \theta = \frac{a^2}{2b} \]