In this article, we will see how to find constant angular acceleration along with examples and solve some problems related to constant angular acceleration.
If the tangential acceleration of the object in a circular motion is constant then this implies that the angular acceleration of the object is also constant.
Angular Velocity of the Object in an Angular Motion
Consider an object accelerating in the circular motion with tangential velocity vt due to the radial displacement ‘s’ in time ‘t’ making an angle ‘θ’.
Hence, the tangential velocity of the object in a angular motion will be
The displacement ‘s’ is the arc length of the circle and is equal to the angle θ made due to the displacement and the radius of the circle given by the equation
Substituting this in the eqn(1), we have
The change in the angle with respect to time gives the angular velocity due to the rotational motion of the object. The angular velocity is represented as
Hence, the previous equation becomes,
The tangential velocity is directly related to the angular velocity by the relation given above.
How to find Angular Acceleration?
The angular acceleration is the change in the angular velocity with time formulated as below
Where α is the angular acceleration of the object.
ω2 is the final angular velocity
ω1 is the initial angular velocity
By calculating the difference between the final and the initial angular velocities of the object we can find the angular acceleration of the object.
Read more on Angular Acceleration.
Relation between the Angular Acceleration and Tangential Acceleration
The tangential acceleration is the variation in the radial velocity with respect to time, given by the relation
at= dvt/dt —(5)
Using equation No. (3), we get
From eqn(4), we have
The angular acceleration of the object will be constant if the tangential acceleration of the object will be constant.
Equation (6) shows the relationship between the tangential acceleration and the angular acceleration of the object and is independent of time.
Problem 1: Consider a pulley of radius 15 cms. The linear acceleration of the pulley is 2m/s2, then find the angular acceleration of the pulley.
The linear acceleration of the pulley at=2m/s2
α =2/0.15=13.33 rad/s2
Hence, the angular acceleration of the pulley is 13.33 rad/s2.
How to find Constant Angular Acceleration?
For an angular acceleration to be constant the angular velocity must be increasing or decreasing with the constant rate.
By measuring the angular velocity that has varied with changing the direction of acceleration of the object in a circular path, we can calculate the angular acceleration of the object.
From the eqn (4), we can say that,
If ω2 > ω1, then we have positive angular acceleration. It means the speed of the object in the angular motion is increasing constantly.
If ω2 < ω1, then we have negative angular acceleration which means that the object is decelerating and the angular velocity of the object is decreasing with time.
If ω2 = ω1, then there is no acceleration of the object in the angular motion. This condition is not possible because the direction of the motion of the object keeps on changing as the direction of the radial velocity changes with respect to the angular velocity.
Problem 2: Consider a wheel of radius 30cms rotating at an angular speed of 2 rad/s which increases to 5 rad/s in 20 seconds. Then calculate the angular acceleration of the wheel.
Given: ω1=2 rad/s
The angular acceleration of the object is 0.15 rad/s2.
Problem 3:Consider a spinning top rotating decelerating at a rate of 2rad/s2 in every 2seconds, the final angular velocity of the object was 0.5 rad/s then find the initial angular velocity of the spinning top.
Given: α= 2rad/s2
Since the object is decelerating, the angular acceleration is negative.
Hence, the initial velocity of the spinning top was 4.5 rad/s.
Angular Acceleration of the Object in a Centripetal Motion
Consider an object of mass ‘m’ moving in a circular path of radius ‘r’. The centripetal force which is responsible for keeping the object in a circular motion is acting inward towards the center of the circle.
This implies the angular acceleration of the object in a centripetal motion is directly proportional to the square of the angular velocity of the object and will be constant if the ω is constant during the process.
Graph of Constant Angular Acceleration
For angular acceleration to be constant the rate of change of angular velocity should be constant.
The above graph shows that the a ngular velocity of the object increases linearly with time, hence the slope of the graph is positive which is equal to the angular acceleration.
If the angular velocity of the object increases at a constant rate with time then the angular acceleration of the object will remain constant for all the time intervals.
Frequently Asked Questions
Q1. A car covers 100m in 30 seconds. The radius of the tyres of a car is 20cms, then calculate the angular of tyres.
Given: r=20cms=0.2m, d=100m, t=30sec
The circumference of the tyre = 2πr
=2 x 3.14 x 0.2=1.256m
The number of rotation of a tyre on traveling 100m is =100/1.256=79.68 rad
Hence, the angular velocity of the tires is
ω =No. of rotations/Time=79.68 rad/30sec=2.656 rad/s
The angular velocity of the tyres is 2.6 rad/s.
Q2. Calculate the angular acceleration of the object at time t=15 sec, if the angular velocity of the object varies from 0.5rad/s to 4 rad/s?
Given: The initial angular velocity ωi =0.5 rad/s
The final angular velocity ωf =4 rad/s
Integrating the above equation,
Hence, the angular acceleration of the object is 0.23 rad/s2.