Answer:
The average angular acceleration is 0.05 radians per square second.
Explanation:
Let suppose that Ferris wheel accelerates at constant rate, the angular acceleration as a function of change in angular position and the squared final and initial angular velocities can be clear from the following expression:
[tex]\omega^{2} = \omega_{o}^{2} + 2 \cdot \alpha\cdot (\theta-\theta_{o})[/tex]
Where:
[tex]\omega_{o}[/tex], [tex]\omega[/tex] - Initial and final angular velocities, measured in radians per second.
[tex]\alpha[/tex] - Angular acceleration, measured in radians per square second.
[tex]\theta_{o}[/tex], [tex]\theta[/tex] - Initial and final angular position, measured in radians.
Then,
[tex]\alpha = \frac{\omega^{2}-\omega_{o}^{2}}{2\cdot (\theta-\theta_{o})}[/tex]
Given that [tex]\omega_{o} = 0\,\frac{rad}{s}[/tex], [tex]\omega = 0.70\,\frac{rad}{s}[/tex] and [tex]\theta-\theta_{o} = 4.9\,rad[/tex], the angular acceleration is:
[tex]\alpha = \frac{\left(0.70\,\frac{rad}{s} \right)^{2}-\left(0\,\frac{rad}{s} \right)^{2}}{2\cdot \left(4.9\,rad\right)}[/tex]
[tex]\alpha = 0.05\,\frac{rad}{s^{2}}[/tex]
Now, the time needed to accelerate the Ferris wheel uniformly is described by this kinematic equation:
[tex]\omega = \omega_{o} + \alpha \cdot t[/tex]
Where [tex]t[/tex] is the time measured in seconds.
The time is cleared and obtain after replacing every value:
[tex]t = \frac{\omega-\omega_{o}}{\alpha}[/tex]
If [tex]\omega_{o} = 0\,\frac{rad}{s}[/tex], [tex]\omega = 0.70\,\frac{rad}{s}[/tex] and [tex]\alpha = 0.05\,\frac{rad}{s^{2}}[/tex], the required time is:
[tex]t = \frac{0.70\,\frac{rad}{s} - 0\,\frac{rad}{s} }{0.05\,\frac{rad}{s^{2}} }[/tex]
[tex]t = 14\,s[/tex]
Average angular acceleration is obtained by dividing the difference between final and initial angular velocities by the time found in the previous step. That is:
[tex]\bar \alpha = \frac{\omega-\omega_{o}}{t}[/tex]
If [tex]\omega_{o} = 0\,\frac{rad}{s}[/tex], [tex]\omega = 0.70\,\frac{rad}{s}[/tex] and [tex]t = 14\,s[/tex], the average angular acceleration is:
[tex]\bar \alpha = \frac{0.70\,\frac{rad}{s} - 0\,\frac{rad}{s} }{14\,s}[/tex]
[tex]\bar \alpha = 0.05\,\frac{rad}{s^{2}}[/tex]
The average angular acceleration is 0.05 radians per square second.
