Angular acceleration is defined as the rate of change of angular velocity with respect to time for a rotating object.
- It is a pseudovector quantity that acts along the axis of rotation (following the right-hand rule).
- The SI unit of angular acceleration is rad/s².
It can be calculated using the following relations:
\alpha = \frac{d\omega}{dt}
\alpha = \frac{\omega_2 - \omega_1}{t_2 - t_1} where ω2 is the final angular velocity, ω1 is the initial angular velocity, t1 is the initial time, and t2 is the final time.
Derivation
Consider an object moving in a circular path of radius r with angular velocity ω\omegaω at time t. Angular acceleration is defined as the rate of change of angular velocity with respect to time:
\alpha = \frac{d\omega}{dt} ....... (1)Angular velocity is defined as the rate of change of angular displacement:
\omega = \frac{d\theta}{dt} Substituting this expression into the equation for angular acceleration:
\alpha = \frac{d}{dt}\left(\frac{d\theta}{dt}\right)
\alpha = \frac{d^2 \theta}{dt^2} Thus, angular acceleration is the second derivative of angular displacement with respect to time.
Solved Problems
Example 1: Calculate the angular acceleration of an object if its angular velocity changes at the rate of 50 rad/s for 5 seconds.
Solution:
dω = 50
dt = 5
Using the formula we have,
α = dω/dt
= 50/5
= 10 rad/s2
Example 2: Calculate the angular acceleration of an object if its angular velocity changes at the rate of 90 rad/s for 4 seconds.
Solution:
dω = 90
dt = 4
Using the formula we have,
α = dω/dt
= 90/4
= 22.5 rad/s2
Example 3: Calculate the angular velocity of an object if its angular acceleration is 30 rad/s2 for 7 seconds.
Solution:
α = 30
dt = 7
Using the formula we have,
α = dω/dt
dω = α dt
dω = 30 (7)
dω = 210 rad/s
Example 4: Calculate the angular velocity of an object if its angular acceleration is 16 rad/s2 for 3 seconds.
Solution:
α = 16
dt = 3
Using the formula we have,
α = dω/dt
dω = α dt
dω = 16 (3)
dω = 48 rad/s
Example 5: Calculate the time taken by an object if its angular velocity is 46 rad/s and acceleration is 23 rad/s2.
Solution:
α = 23
dω = 46
Using the formula we have,
α = dω/dt
dt = dω/α
dt = 46/23
dt = 2 s
Unsolved Problems
Question 1: An object’s angular velocity changes by 80 rad/s in 8 seconds. Find its angular acceleration.
Question 2: A rotating body has an angular acceleration of 12 rad/s² for 5 seconds. Find the change in angular velocity.
Question 3: An object’s angular velocity increases from 10 rad/s to 70 rad/s in 6 seconds. Find its angular acceleration.
Question 4: A wheel has an angular acceleration of 20 rad/s² and its angular velocity changes by 100 rad/s. Find the time taken.
Question 5: An object starts from rest and reaches an angular velocity of 60 rad/s in 4 seconds. Find its angular acceleration.