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 find the angle through which a rigid body rotates before it comes to rest, given that its angular velocity is a function of time, we can follow these steps:

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

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

### Step 3: Relate angular displacement to angular velocity
The angular displacement \( \theta \) can be expressed in terms of angular velocity:
\[ d\theta = \omega dt \]
Substituting the expression for \( \omega \):
\[ d\theta = (a - bt) dt \]

### Step 4: Integrate to find the total angle
To find the total angle \( \theta \) through which the body rotates before coming to rest, we need to integrate \( d\theta \) from \( t = 0 \) to \( t = \frac{a}{b} \):
\[ \theta = \int_0^{\frac{a}{b}} (a - bt) dt \]

### Step 5: Perform the integration
We can break this integral into two parts:
\[ \theta = \int_0^{\frac{a}{b}} a \, dt - \int_0^{\frac{a}{b}} bt \, dt \]
Calculating each integral:
1. The first integral:
\[ \int_0^{\frac{a}{b}} a \, dt = a \left[ t \right]_0^{\frac{a}{b}} = a \cdot \frac{a}{b} = \frac{a^2}{b} \]
2. The second integral:
\[ \int_0^{\frac{a}{b}} bt \, dt = b \left[ \frac{t^2}{2} \right]_0^{\frac{a}{b}} = b \cdot \frac{1}{2} \left( \frac{a^2}{b^2} \right) = \frac{a^2}{2b} \]

### Step 6: Combine the results
Now, substituting back into the equation for \( \theta \):
\[ \theta = \frac{a^2}{b} - \frac{a^2}{2b} = \frac{a^2}{2b} \]

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

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