Angular impulse, definition, mathematical expression, practice problems, FAQs

Suppose you throw a ball at a door, while the ball bounces back, the door experiences a torque for a very short moment. This torque applied while the ball was in contact with the door gave it an impulse. This impulse is responsible for change in angular momentum of the torque.

Table of contents:

Angular Impulse
Practice problems
Faqs

Angular momentum

Angular momentum is a mathematical quantity that quantifies the effect of torque over a body for a short interval.

Units of angular impulse is and its dimension is .

The change in the angular momentum suffered by a system due to the action of a torque for an infinitesimal amount of time is the angular impulse given by the torque to the system.

The angular impulse applied to an object in a given time interval is the area under torque vs time (τ - t) graph with the time axis in the same time interval.

Area

Practice problem

Q.Two cylinders of radii and that have the moments of inertia, and , about their respective axes, rotate about their axes with angular speeds and , respectively. The cylinders are moved closer to touch each other by keeping the axes parallel. The slipping of the cylinders ceases due to friction between them although they were slipping over each other at first. What will be the cylinders angular speeds after the slipping ceases?

Let the angular velocities of the two cylinders be and before the slipping and and when the slipping ceases.

A kinetic friction acts on the cylinders when there is slipping between the cylinders because they have different velocities at contact. Due to this kinetic friction, the velocities of the point of contact changes till they become equal.

Thus, when the slipping ceases,

The angular impulse due to the friction force about the hinge is as follows:

Taking the anticlockwise direction as positive.

For cylinder 1,

Dividing equation by , we get the following,

However from equation , we get

Substituting in equation , we get

and

Q. A pulley is rotated about its axis by a force (where is measured in seconds) applied tangentially. Find the magnitude of angular impulse [in ] on the pulley in the initial . Given radius of pulley .

A. As we know,

Angular Impulse

Where

where is the perpendicular distance of the force vector from the axis of rotation.

i.e (since force acts tangentially,

Hence,

Q. A rod of mass and length is lying in the horizontal plane and pivoted about its one end. The rod is rotating about its pivoted end initially (axis perpendicular to horizontal plane) with an angular velocity . Suddenly, an angular impulse is given to the rod, because of which its angular velocity becomes . Find the magnitude of angular impulse .

A. Given,

Mass of rod,

Length of rod,

MOI of rod about its end,

As we know from impulse-angular momentum theorem,

where, = final angular momentum

initial angular momentum

Final angular momentum

Therefore,

Q. A uniform solid sphere is placed on a smooth horizontal surface. At a height above the centre line, an impulse is given horizontally to the sphere . Assume the mass and radius of the sphere being and respectively.

Find angular velocity of the sphere and linear velocity of the centre of mass of the sphere after the impulse.

A. Assume that just after the impulse, linear velocity of the sphere is and angular velocity is .

As we know,

Impulse = change in linear momentum

(, because initially the sphere is at rest.)

is the linear velocity of the COM

Angular impulse = change in angular momentum

, because initially the sphere is at rest.

Therefore, is the angular velocity of the sphere after the impulse

FAQs

Q. What would happen to angular momentum, if angular impulse on a particle is zero?
A. Angular momentum will remain conserved.

Q. Is angular impulse a scalar quantity or vector quantity?
A. Angular impulse is a vector quantity.

Q. Why is angular impulse important in real life?
A. Torque in real life is often not constant and tends to build up from zero over time and may vary depending on many factors. To find the overall effect of all these forces directly is quite cumbersome. Therefore by finding the area under the curve we can easily quantify the overall effect of torque over time.

Q. What is the relation between angular impulse and impulse?
A. Angular impulse is the rotational analogue of impulse.