Angular acceleration, types, relation with tangential acceleration, practice problems, FAQs

You might have observed that when you switch ON the fan, it starts from rest and continuously increases its velocity till it comes to the particular speed you chose. This means, similar to the acceleration that occurs in linear motion, rotational and circular motion also has some acceleration. This acceleration is known as angular acceleration. Let’s see more about this here!

Table of contents

Angular acceleration
Relationship between Tangential Acceleration and Angular Acceleration
Average angular acceleration
Instantaneous angular acceleration:
Total Acceleration in Circular Motion
Comparison of uniform circular motion (UCM) with non-uniform circular motion (NUCM)
Analogy between Linear and Angular Kinematics
Equations of motion in non-uniform circular motion
Practice problems
FAQs

Angular acceleration

It is the rate of change of angular velocity.

Relationship between Tangential Acceleration and Angular Acceleration

We know that,

Where,
is the linear velocity
is the angular velocity
is the position vector of the point under consideration from the axis of rotation
In scalar form we can write,

Differentiating both sides with respect to t,

It is to be noted that, in a non-uniform circular motion, both angular and tangential acceleration exist.

Average angular acceleration

If and are the instantaneous angular velocities at times and , respectively, then the average angular acceleration is,

SI unit of angular acceleration is

Instantaneous angular acceleration

It is the instantaneous rate of change of angular velocity with time.

SI unit:

Relationship between Angular Acceleration and Angular Velocity

Let be the angular acceleration and be the angular velocity of a particle in the circular motion at any instant of time .

Let be the angular position of the particle at time t and it covers an angular displacement in time .

We know that,

Total Acceleration in Circular Motion

Consider a particle moving in a circular motion.

At any instant , let the velocity of the particle be and the net acceleration be

We know,

But
So,

Thus, total acceleration is the vector sum of tangential acceleration and centripetal acceleration.

Where is the angle made by the resultant acceleration with the centripetal acceleration.

Comparison of uniform circular motion(UCM) with non-uniform circular motion(NUCM)

Parameter	UCM	NUCM
Angular speed	Constant	Variable
Angular velocity	Constant	Variable
Angular acceleration	Zero	Non-zero
Tangential acceleration	Zero	Non-zero
Centripetal acceleration	Non-zero	Non-zero

Analogy between Linear and Angular Kinematics

Parameter	Linear kinematics	Angular kinematics
Position/Angle	Position (_$\vec{X}$)	Angle (_$\vec{θ}$)
Velocity	$\vec{v} = \frac{d \vec{x}}{d t}$	$\vec{ω} = \frac{d \vec{θ}}{d t}$
Acceleration	$\vec{a} = \frac{d \vec{v}}{d t} = \frac{d^{2} \vec{x}}{d t^{2}}$	$\vec{α} = \frac{d \vec{ω}}{d t} = \frac{d^{2} \vec{θ}}{d t^{2}}$

Equations of motion in non-uniform circular motion

For applying equations of motion in non-uniform circular motion, we can do it only for a constant angular and tangential acceleration.

The equations of motion in angular kinematics are:

In scalar and simple form,

Angular kinematics	Linear kinematics
$ω = ω_{o} + α t$	$v = u + a t$
$Δ θ = ω_{o} t + \frac{1}{2} α t^{2}$	$s = u t + \frac{1}{2} a t^{2}$
$ω^{2} = {ω_{o}}^{2} + 2 α Δ θ$	$v^{2} = u^{2} + 2 a s$

Practice problems

Q1. A particle moves in a circle of radius of at a speed that increases uniformly. Find the angular acceleration of the particle, if its speed changes from to in .

Ans. Given,

Radius =
Final speed =
Initial speed =
Time taken = 4 s

We know that,

Also tangential acceleration, (where Angular acceleration)

Q2. A pulley wheel of diameter 8 cm has a 5m long cord wrapped around its periphery. Starting from rest, the wheel is given an angular acceleration of.

Through what angle must the wheel turn for the cord to unwind completely?
How long does it take?

Ans. Given,

Radius of wheel () =
Initial angular velocity ()
Angular acceleration ()

This is a case of constant angular acceleration, so we can use the equations of motion.

(a) Since the cord is to unwind completely, the equivalent linear displacement of the wheel is equal to the length of the string.

Now, angular displacement of wheel,

(b) For the time taken, we use the equation,

(we can neglect negative value of t)

Q3. A solid body rotates about a stationary axis with an angular retardation , where 𝜔 is the angular velocity of the body. Find the time after which the body comes to rest if at , the angular velocity of body was .

Ans. Given,

Initial angular velocity =
Angular retardation,
Final angular velocity () = 0
We need to find time (t) after which the body comes to rest.

We know that,

(Negative sign due to retardation)

Integrating both sides with limits,

Q4. A solid body rotates about a stationary axis with an angular retardation , where is the angular velocity of the body at an instant. Find the average angular speed of the body, averaged over the whole time of rotation if at the initial moment of time, its angular velocity was equal to .

Ans. Given,

Angular retardation, where is the angular velocity

Initial angular velocity =

We have,

Average angular velocity,

Now,

Integrating both sides with limits,

Also angular acceleration,

Integrating both sides with limits,

Substituting and equation in equation,

FAQs

Q1. What are the dimensions of angular acceleration?

Ans. The angular acceleration is the rate of change of angular velocity with time.

So it is the double derivative with respect to time over the angular displacement. We know that the angle expressed in terms of radians has no dimension as it is a ratio of arc length to the length of radius. So the dimension of angular acceleration is .

Q2. What happens to the angular acceleration, if angular velocity becomes constant?

Ans. The angular acceleration becomes zero when angular velocity remains constant. Since , if the angular velocity remains constant, the rate of change will become zero.

Q3. What causes constant angular acceleration?

Ans. Constant angular acceleration is achieved in a uniform circular motion. This happens when there is no torque acting on the body under consideration. We know, torque .

So, angular acceleration becomes zero only when the torque acting is also zero and thus the constant angular acceleration is achieved.

Q4. Why does angular acceleration not change with radius?

Ans. Angular acceleration is the rate of change of angular velocity with time. Angular velocity is the rate of change of angular displacement with time. Angular displacement is measured as the angle subtended by the radius. It does not change with the radial distance from the axis. So the angular velocity and angular acceleration also doesn't change with the radial distance from the axis.