A solid body rotates about a stationary axis so that its angular velocity depends upon the rotation angle `phi ` as `omega - omega_(0) = aphi`, where a and `omega_(0)` are positive constants. Initially that is at t `= 0 , phi = 0 `.

Options

A. `phi = ( omega_(0))/( a ) ( 1 - e^(-at))`
B. `phi = ( omega_(0))/( a ) ( 1 + e^(-at))`
C. `omega = omega_(0) e^(-at)`
D. `omega= omega_(0) e^(at)`

Answer

To solve the problem, we need to analyze the relationship between angular velocity (ω), the rotation angle (φ), and time (t) based on the given equation:

\[ \omega - \omega_0 = -a\phi \]

where \(a\) and \(\omega_0\) are positive constants.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The equation can be rearranged to express angular velocity in terms of the rotation angle:
\[ \omega = \omega_0 - a\phi \]

2. **Relating Angular Velocity and Angle**:
The angular velocity is defined as the rate of change of the angle with respect to time:
\[ \frac{d\phi}{dt} = \omega \]
Substituting for \(\omega\):
\[ \frac{d\phi}{dt} = \omega_0 - a\phi \]

3. **Separating Variables**:
We can rearrange this equation to separate variables:
\[ \frac{d\phi}{\omega_0 - a\phi} = dt \]

4. **Integrating Both Sides**:
We will integrate both sides. The left side will be integrated from \(\phi = 0\) to \(\phi\), and the right side from \(t = 0\) to \(t\):
\[ \int_0^{\phi} \frac{d\phi}{\omega_0 - a\phi} = \int_0^{t} dt \]

5. **Calculating the Left Integral**:
The left integral can be solved using the natural logarithm:
\[ -\frac{1}{a} \ln|\omega_0 - a\phi| \bigg|_0^{\phi} = t \]
Evaluating the limits:
\[ -\frac{1}{a} \left( \ln(\omega_0 - a\phi) - \ln(\omega_0) \right) = t \]
This simplifies to:
\[ -\frac{1}{a} \ln\left(\frac{\omega_0 - a\phi}{\omega_0}\right) = t \]

6. **Exponentiating Both Sides**:
To eliminate the logarithm, we exponentiate both sides:
\[ \frac{\omega_0 - a\phi}{\omega_0} = e^{-at} \]
Rearranging gives:
\[ \omega_0 - a\phi = \omega_0 e^{-at} \]

7. **Solving for φ**:
Now, we can solve for \(\phi\):
\[ a\phi = \omega_0 - \omega_0 e^{-at} \]
\[ \phi = \frac{\omega_0(1 - e^{-at})}{a} \]

### Final Expression:
Thus, the expression for the rotation angle \(\phi\) in terms of time \(t\) is:
\[ \phi = \frac{\omega_0}{a}(1 - e^{-at}) \]