Angular Acceleration And Angular Momentum: Facts To Consider

The terms, angular acceleration and angular momentum define the behavior of the motion of the rotating objects.

The angular momentum is a quantity that keeps the object spinning around about its axis of inertia, and the angular acceleration is due to the applied torque on the body. Both the quantities determine the magnitude and the direction of the spinning object.

Is angular momentum the same as angular acceleration?

The angular momentum and angular acceleration are quite different terms though they determine the components of the same motion of the object.

The angular momentum is a quantity that keeps the object rotating about an axis, and the angular acceleration of the object tells about the speed of rotation of the object. The momentum of the object rotating about an axis varies as the torque is exerted on its body.

The torque felt on the body displaces the object in the linear and the rotational motion. The momentum of the object is conserved in the a case of the spinning object having the angular displacement.

Relation between Angular Acceleration and Angular Momentum

The angular acceleration of the object is given by the relation:

gif

Here, α is the angular acceleration,

ω is angular velocity, and

t is a time

The angular acceleration is the rate of variation in the angular velocity in every interval of time and is measured in terms of rad/s2

The angular momentum is the angular displacement occurred by maintaining the momentum of the object. The momentum here is the angular velocity of the object times the moment of inertia of the body that it possesses.

The angular momentum of the object is given by the expression:

gif

Here, L is the angular momentum of the object,

m is the mass of the object,

v is a velocity and

r is a displacement of the object.

The torque exerted on the object that keeps it in an angular motion is given as:

Using equation (1) in the equation (2), we get,

gif

Now, we know that the change in angular velocity is nothing but the angular acceleration of the object, hence we can simply rewrite the above equation as:

gif

Equating equation (2) & (3), we get,

gif

This equation gives the relation between the angular momentum and the angular acceleration of the body in a rotational motion.

According to the above relation, we can state that

“The change in the angular momentum of the rotating body is directly proportional to the angular acceleration and its moment of inertia.”

How to find angular acceleration from angular momentum?

The change in momentum is due to the torque experienced on the body.

Hence if we know the amount of torque exerted, we can find the rate of change in the angular velocity of the object. If more is the inertia of the object, the more force is required to apply the torque on the body.

angular acceleration and angular momentum
Hand spinner on giving a torque; Image Credit: pixabay

The variation in the angular momentum is seen when the torque is exerted on the object by applying a force that increases the angular velocity of the object. The increase in the angular velocity is inevitable for the change in the angular acceleration of the object.

Let us see how to calculate the angular acceleration from the angular momentum of the object taking one simple problem.

The solid cylinder has a mass of 5 kg and a radius of 18 cm moving with an angular momentum 1250 kg.m2/s. What is the angular acceleration of the cylinder if the angular moment changes to 1875 kg.m2/s after 2 min?

Given: m =5kg

r =18cm =0.18m

s

t =2 min =120 sec

The moment of inertia of a solid cylinder is,

The relation between the angular momentum and angular acceleration is given as

s%5E2

Hence, the angular acceleration of the solid cylinder is 64.20/s2

Angular Momentum v/s Angular Acceleration

Characteristics Angular Momentum Angular Acceleration
Definition It is a product of the moment of inertia and angular velocity of the object. It is the variation in the angular velocity of the spinning object.
Conservation of Momentum The object spins conserving the total momentum of its body. The angular acceleration keeps on varying depending upon the torque experienced on the object of the rotating object.
Units It is measured in terms of kg.m2/s. It is measured as rad/s2.
Mass It relies upon the mass and configuration of the body. It is independent of the mass of the object.
Configuration It depends upon the shape, size, distribution of mass, and the axis of rotation of the body. It depends on the change in velocity and the external force applied.
Work It keeps the object spinning about its axis of rotation. It accelerates the angular speed of the object.
Direction Angular momentum is not in a direction of the angular velocity. Angular acceleration is in a direction with the angular velocity.
Angular Velocity It can be changed by keeping angular velocity constant. It can be changed by changing the angular velocity.

What is the angular acceleration of the ball having a mass of 500 grams and a radius of 20 cm on experiencing a torque of 25N.m?

Given: m =500 grams =0.5 kg

r =20 cm= 0.2 m

τ=3 N.m

The moment of inertia of the ball is

We have,

s%5E2

The angular acceleration is 230/s2

Conclusion

The angular acceleration and angular momentum of the object both are dependent quantities. The change in the speed of the spinning object determines the angular acceleration which in turn is related to the changing angular momentum.

Also Read:

Leave a Comment