# Angular Velocity vs Angular Acceleration: 3 Important Facts

The concepts angular velocity and angular acceleration are the most familiar concept to explain how fast a body can change its position and how rapidly it travels along the circular path.

When you rotate a ball in the circular orbit, it rotates at a certain angle with a certain velocity—this velocity results in acceleration. Let us discuss angular velocity vs angular acceleration in this post.

Angular velocity:

We have discussed the angular velocity in the previous article.

The differential change of displacement of an object rotating along the circular orbit at an angle ‘θ’, with the time is called the angular velocity.

The formula for angular velocity is,

Angular acceleration:

The concept of angular acceleration is similar to linear acceleration.

The rate of change of velocity of an object rotating at angle ‘θ’ in a circular orbit with time is called angular acceleration.

It is denoted by the Greek letter ‘α’.

If a body is moving in a circular path with velocity ωi initially and it changes its velocity to ωf, then the acceleration of the moving body is given by

But ∆ω = ωf -ω0

Angular acceleration is given by the difference between angular velocities of initial and final velocity.

## Angular Velocity Vs Angular Acceleration

The comparison between the angular velocity and angular acceleration is given in the below table, which may help you to understand.

## Facts related to angular velocity and angular acceleration:

• In a two-dimensional space, the angular acceleration can change its sign or can be inverted with the coordinates. It is called pseudo-scalar quantity.
• When the velocity of the rotating object increases, the acceleration is positive.

When you switch on a fan, it starts rotating from zero and keeps on increasing whenever you turn the nob to get more air. In that case, acceleration is positive.

• When the velocity decreases while rotating, the angular acceleration will become negative.

Whenever you turn the nob of a fan in an anti-clockwise direction to lower the speed, you can observe negative acceleration.

• In case of increase in the angular velocity, the angular acceleration and the velocity will be in the same direction.
• The angular acceleration will act opposite to the angular velocity whenever there is a decrease in the velocity.
• In a uniform circular orbit, the velocity vector exhibit constant magnitude.
• Angular acceleration will become zero in a uniform circular orbit.
• Angular acceleration decreases in the rotational path of maximum radius.

## A car is moving in a circular path. Initially, the angular velocity of the car is 26km/hr, and after 34 min, it increases its speed to 49km/hr. Calculate the angular acceleration of the car.

Solution:

The initial velocity ωi = 26km/hr

The change in velocity ωf = 49km/hr

Time is 34 min = 0.56 hr

The angular acceleration is

## The wheel of a cycle is rotating with an angular acceleration of 12rad/sec2 in 3seconds. Calculate the angular velocity.

Solution:

The Angular acceleration of the wheel is 12 rad/sec2

Time taken to accelerate is 3 seconds

The angular acceleration is given by

Then the velocity can be written in terms of angular acceleration as

∆ω = α.∆t

∆ω = 12 × 3

## A disc of a radius of 12cm is rotating in a circular path with an angle of 35°. The time taken to complete rotation is 12 seconds. Calculate the angular velocity and hence find out the angular acceleration if it increases its velocity to 4 rad/sec for the same 12 seconds.

Solution:

The angle of rotation = ∆θ = 35°

Time taken for one complete rotation ∆t = 12 seconds

Angular velocity is given by the formula

The angular acceleration is given by

The velocity is changed to 4 rad/sec for the same time interval so that the angular acceleration is given by;

## A tire is rotating with an acceleration of 65 rad/sec. Its change in velocity is given by 92 rad/sec2. Calculate the time taken by the tire to gain the given acceleration?

Solution:

The angular velocity = 92 rad/sec

The angular acceleration is

Rearranging the above equation, we can calculate the time as,

∆t = 1.41sec.

## Does the angular velocity depend on the mass of the rotating object?

Yes, the angular velocity inversely depends on the mass.

When a freely rotating body of a certain mass is supposed to exert some velocity, if the mass is more, then the velocity decreases.

## Does the radius affect theangular acceleration?

Suppose angular acceleration is maximum, the radius of the rotational path matters.

Greater the radius of the orbit, the object’s attraction towards the center becomes less. This results in the decrease of the velocity and hence the acceleration.

## Why does the angular acceleration is zero in a uniform circular orbit?

Angular acceleration refers to a change in the Angular velocity; either the magnitude has to be changed or the speed.

In a uniform circular orbit, the velocity remains constant throughout. The neither magnitude nor the radius changes. This shows that there will be no acceleration produced.

## Does the tangential acceleration and angular acceleration are same?

There are two types of acceleration;

• Linear acceleration
• Angular acceleration

When a body is accelerating in the circular orbit, it is said to be angular acceleration. This angular acceleration is further divided as