Angular acceleration is defined as the rate at which angular velocity changes over time. It is a pseudovector in three dimensions. The SI unit of measurement is radians/second squared (rad/s2). Moreover, we commonly represent it by the Greek letter alpha (α). Learn about the angular acceleration formula here.
What is Angular Acceleration?
Angular acceleration is a measure of how quickly an object’s rotational velocity changes with time. It is a vector quantity, meaning it has both magnitude and direction. In simpler terms, angular acceleration tells us how fast the rate of rotation of an object is increasing or decreasing. Angular acceleration occurs when a torque (rotational force) is applied to an object, causing it to change its state of rotational motion.
The unit of angular acceleration in the International System of Units (SI) is radians per second squared (rad/s²).
Angular Acceleration Formula
Angular Acceleration is usually expressed in radians per second whole square. Thus,
α = dω / dt
where,
α is the angular acceleration
ω is the angular velocity
t is the time taken by the object
If angular displacement θ is given, then the angular acceleration is calculated as,
α = d2θ/dt2
Angular Acceleration Types
There are mainly two types of angular acceleration:
- Spin Angular Acceleration.
- Orbital angular acceleration
These two indicate the temporal rate of change in spin angular velocity and orbital angular velocity, respectively. Unlike linear acceleration, rotational acceleration does not have to be generated by the following external torque. A figure skater, for example, can quickly speed up his or her spin (and so acquire angular acceleration) by squeezing his or her arms inwards, with no personal torque required.
Angular Acceleration Formula Derivation
Suppose an object is doing circular motion with a linear velocity v, angular velocity ω on a circular path of radius r in time t. Now, we know the angular acceleration of an object is the first derivative of its angular velocity with respect to time. So we get,
α = dω/dt ……. (1)
Also, we know that the angular velocity of an object is the first derivative of its radius with respect to time.
ω = dθ/dt… (2)
Substituting (2) in (1) we get,
α = d(dθ/dt)/dt
α = d2θ/dt2
This derives the formula for angular acceleration.
Angular Acceleration Solved Examples
Example 1: Find the angular acceleration of an object if its angular velocity changes at the rate of 100 rad/s for 10 seconds.
Solution:
dω = 100
dt = 10
Using the formula we have,
α = 100/10
α = 10 rad/s2
Example 2: Calculate the angular acceleration of an object if its angular velocity changes at the rate of 124 rad/s for 4 seconds.
Solution:
dω = 124
dt = 4
Using the formula we have,
α = dω/dt
α = 124/4
α = 31 rad/s2
Example 3: A disk with a radius of 0.2 meters is initially at rest. A constant torque is applied, causing the disk to reach an angular velocity of 10 rad/s in 5 seconds. Calculate the angular acceleration.
Solution:
Given Data:
Initial angular velocity (\(\omega_0\)) = 0 rad/s
Final angular velocity (\(\omega\)) = 10 rad/s
Time (\(t\)) = 5 seconds
Formula for Angular Acceleration:
\[\alpha = \frac{\Delta \omega}{\Delta t}\]
Substitute the Values:
\[\alpha = \frac{10 \, \text{rad/s} – 0 \, \text{rad/s}}{5 \, \text{s}} = \frac{10 \, \text{rad/s}}{5 \, \text{s}} = 2 \, \text{rad/s}^2\]
Answer: The angular acceleration of the disk is \(2 \, \text{rad/s}^2\).