Question

A rotating wheel changes its angular speed from 1800 rpm to 3000rpm in 20s.what is the angular acceleration assuming it to be uniform?
A. $60\pi ra{{d}^{-2}}$
B. $90\pi ra{{d}^{-2}}$
C. $2\pi ra{{d}^{-2}}$
D, $40\pi ra{{d}^{-2}}$

Answer 1

The question includes concept from rotational motion.
The steps to be followed to solve this question are
Find the initial and final angular velocity from the given value of angular speed.
Find the difference between the angular velocity found in the first step.
Divide the difference by the given time as we know angular acceleration is change in angular velocity per unit time.

Complete answer:
The vector direction of angular acceleration is perpendicular to the plane where the rotation takes place. Increase in angular velocity clockwise, then the angular acceleration velocity points away from the observer. If the increase in angular velocity counterclockwise, then the vector of angular acceleration points toward the viewer.
The amount of angle covered per square of time is measured in radian per second squared. The unit is degrees/square of time.
For finding angular acceleration from the given data, we need to convert the frequency into rotations per second then we can find the angular acceleration of the wheel.
Let us name initial angular velocity ${{w}_{1}}=2\pi {{f}_{1}}$where ${{f}_{1}}$ is initial frequency.
And final angular velocity ${{w}_{2}}=2\pi {{f}_{2}}$where ${{f}_{2}}$ is initial frequency.
The given frequencies are in rotations per minute, we will convert them in rotations per second.
${{f}_{1}}$=$1800rpm$=$\dfrac{1800}{60}$=$30$ rotations per second
${{f}_{2}}$=$3000rpm$=$\dfrac{3000}{60}$=$50$ rotations per second
Time in which the frequency changes = 20 seconds
Angular acceleration $\left( \alpha \right)$=$\dfrac{{{w}_{1}}-{{w}_{2}}}{time}$
$\Rightarrow \dfrac{2\pi \left( {{f}_{1}}-{{f}_{2}} \right)}{time}$
Putting values
$\Rightarrow \dfrac{2\pi \times 20}{20}ra{{d}^{-2}}$
$\Rightarrow 2\pi \,ra{{d}^{-2}}$

This is the angular acceleration which is option C.

Note:
Angular acceleration is a vector quantity.
Angular momentum, property characterizing the rotatory inertia of an object or system of objects which are in motion about an axis that may or may not pass through the object or system.
The magnitude of the angular momentum of an orbiting object is equal to its linear momentum times the perpendicular distance r from the center of rotation to a line drawn in the direction of its instantaneous motion and passing through the object’s centre of gravity.