10.2 Rotation with constant angular acceleration  (Page 3/5)

In the preceding example, we considered a fishing reel with a positive angular acceleration. Now let us consider what happens with a negative angular acceleration.

Angular acceleration of a propeller

Strategy

1. Since the angular velocity varies linearly with time, we know that the angular acceleration is constant and does not depend on the time variable. The angular acceleration is the slope of the angular velocity vs. time graph, $\alpha =\frac{d\omega }{dt}$ . To calculate the slope, we read directly from [link] , and see that ${\omega }_0=30\text{rad/s}$ at $t=0\text{s}$ and ${\omega }_{\text{f}}=0\text{rad/s}$ at $t=5\text{s}$ . Then, we can verify the result using $\omega ={\omega }_0+\alpha t$ .
2. We use the equation $\omega =\frac{d\theta }{dt};$ since the time derivative of the angle is the angular velocity, we can find the angular displacement by integrating the angular velocity, which from the figure means taking the area under the angular velocity graph. In other words:
   
   ${\displaystyle \underset{{\theta }_0}{\overset{{\theta }_{\text{f}}}{\int }}d\theta ={\theta }_{\text{f}}-{\theta }_0=}{\displaystyle \underset{t_0}{\overset{t_{\text{f}}}{\int }}\omega (t)dt}.$
   
   Then we use the kinematic equations for constant acceleration to verify the result.