Question 308: Work, Energy and Power & Circular Motion

Question

A particle starting from rest moves along the circumference of a circle of radius $$r$$ with angular acceleration $$\alpha$$. The magnitude of the average velocity, in the time it completes the small angular displacement $$	heta$$ is

Accepted Answer

To find the magnitude of the average velocity of a particle moving along a circular path with angular acceleration, we first consider the kinematics of rotational motion.
The angular displacement, $$	heta$$, is given by the rotational kinematics equation for a particle starting from rest:
$$ 	heta = \frac{1}{2} \alpha t^2 $$
Solving for the time $$t$$:
$$ t = \sqrt{\frac{2	heta}{\alpha}} $$
The tangential velocity $$v$$ at any time $$t$$ for a particle traveling along the circumference of a circle with angular velocity $$\omega$$ is related by:
$$ v = r\omega $$
Since the angular velocity $$\omega$$ is related to angular acceleration $$\alpha$$ by:
$$ \omega = \alpha t $$
Substituting $$t = \sqrt{\frac{2	heta}{\alpha}}$$:
$$ \omega = \alpha \sqrt{\frac{2	heta}{\alpha}} = \sqrt{2\alpha	heta} $$
Therefore, the tangential velocity $$v$$ becomes:
$$ v = r \sqrt{2\alpha	heta} $$
Average velocity $$\overline{v}$$ over the time interval during which the particle covers the angular displacement $$	heta$$ is computed as the total arc length traveled divided by the time taken. The arc length $$s$$ is given by:
$$ s = r	heta $$
Thus, the average velocity is:
$$ \overline{v} = \frac{s}{t} = \frac{r	heta}{\sqrt{\frac{2	heta}{\alpha}}}  $$
Simplifying:
$$ \overline{v} = r 	heta \cdot \sqrt{\frac{\alpha}{2	heta}} = r \sqrt{\frac{\alpha 	heta}{2}} $$
Therefore, the correct choice is:
Option D
: $$r\left(\frac{\alpha 	heta}{2}ight)^{\frac{1}{2}}$$

Answer

$$r\left(\frac{2}{\alpha 	heta}ight)^2$$

Answer

$$r\left(\frac{\alpha 	heta}{2}ight)^2$$

Answer

$$ho\left(\frac{\alpha 	heta}{2}ight)$$

Answer

$$r\left(\frac{\alpha 	heta}{2}ight)^{\frac{1}{2}}$$

Work, Energy and Power & Circular Motion

A particle starting from rest moves along the circumference of a circle of radius $r with angular acceleration \alpha. The magnitude of the average velocity, in the time it completes the small angular displacement \theta$ is

Choose the correct answer: