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.
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,
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.
|Angular Velocity||Angular Acceleration|
|The differentiation of angular displacement with time gives the angular velocity.||The second order differentiation of angular displacement time gives the angular acceleration.|
|The unit of angular velocity is radians/second.||The unit of angular acceleration is radians/second2 .|
|The dimensional formula of angular velocity is [M0L0T-1]||The dimensional formula of angular displacement is [M0L0T-2]|
|It has magnitude, but the direction changes with the coordinate axes; hence it is a pseudo vector quantity.||It has magnitude and a specific direction, which remains constant through out the action; hence it is a vector quantity.|
|Radius of the circular orbit does not exhibit any effect on the angular velocity.||The radius of the circular orbit exhibits an inverse effect on the angular acceleration.|
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.
The initial velocity ωi = 26km/hr
The change in velocity ωf = 49km/hr
Time is 34 min = 0.56 hr
The angular acceleration is
α = 2.15 rad/sec2.
The wheel of a cycle is rotating with an angular acceleration of 12rad/sec2 in 3seconds. Calculate the angular velocity.
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
∆ω = 36 rad/sec.
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.
The angle of rotation = ∆θ = 35°
Time taken for one complete rotation ∆t = 12 seconds
Angular velocity is given by the formula
ω = 2.91 rad/sec.
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;
α = 0.090 rad/sec2.
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?
The angular acceleration = 65rad/sec2
The angular velocity = 92 rad/sec
The angular acceleration is
Rearranging the above equation, we can calculate the time as,
∆t = 1.41sec.
Frequently Asked Questions
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 the angular 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
- Tangential acceleration
- Radial acceleration
So tangential acceleration is derived from angular acceleration. It is not the same as angular acceleration.
Can the angular acceleration be negative?
The negative angular acceleration depends on the coordinate axis in which the angular velocity is acting.
When an object is supposed to travel in a circular path with a certain velocity, the change in the velocity gives rise to angular acceleration. If the change in velocity decreases the amount, then the angular acceleration will be negative.