Angular Momentum

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Angular Momentum:

  • The angular momentum of a rigid body about a given axis is defined as the product of its moment of inertia about the given axis and its angular velocity.
  • Its S.I. unit is kg m² s-1 and its dimensions are [M1L2T-1]

Relation Between Angular Momentum and Angular Velocity of a Rigid Rotating Body:

Angular Momentum 01

  • Consider a rigid body rotating about axis passing-through O and perpendicular to the plane of the paper. Consider the infinitesimal element of mass dm at A. Let be its position vector w.r.t. point O. Let   be the linear velocity of the element, then the linear momentum of the element is given by

dpv .dm

Considering magnitude only

dP =  v .dm



  • Then, quantity dL =  r . dp is the magnitude of the angular momentum of the element. Similarly, we can find the angular moment of each and every element in the body.  As all elements are moving in the same direction, resultant angular momentum can be calculated by integrating the above expression.

Angular Momentum 02

This is an expression for the angular momentum of a rotating body.

In vector form,

L = I ω



Principle of Conservation of Angular Momentum:

  • Statement: If the external torque acting on the body or the system is zero, then the total vector angular momentum of a body or of a system remains constant.
  • Explanation: For a rigid body rotating about given axis, torque is given by

τ = I α

Where τ =  torque acting on rotating body

I  =  Moment of Inertia of the body about given axis of rotation

α =   Angular acceleration

Angular Momentum 03



For rigid body moment of inertia about given axis of rotation is always constant.

 

Angular Momentum 04

As external torque acting on the body or the system is zero, τ  =  0

Angular Momentum 05



Therefore, there is no change In angular momentum.

∴ L  =   constant,

but L = Iω

∴ L =   Iω =   constant

I1ω1 = I2ω2 = I3ω3 = ……. = Inωn = constant





  • Above relation indicates that as the moment of inertia decreases angular velocity increases and vice-versa.  This is known as the principle of conservation of angular momentum.
  • Moment of Inertia of body changes if the distribution of mass about the axis of rotation changes.

Examples of Conservation of Angular Momentum:

  • This principle is used by acrobats in the circus, ballet dancers, skaters etc.  By extending or by pulling in the hands, legs. they change the distribution of mass. about the axis of rotation and thus their angular velocity changes by keeping angular momentum constant.  If they, extend their hands or legs the moment of Inertia increases thus angular velocity decreases.  If they pull in their hands or legs the moment of inertia decreases thus angular velocity increases. When diver, want to execute somersault, he pulls in his arms and legs together, so that the moment of inertia decreases and his angular velocity increases.

Kinetic Energy of Rolling Body:

Expression for Kinetic Energy of a Body Rolling on a Horizontal Surface Without Sliding (Slipping):

  • When body rolls, without slipping along a horizontal surface, it has both the translational as well as the rotational Kinetic Energy.  Let rolling body have a mass M and radius R. Let its centre of mass be advancing with a velocity v. Then

Angular Momentum 06

  • If ω is the angular velocity of the body about an axis through the centre of mass. Let I be the M.I. of the body about the axis of rotation. Then

Angular Momentum 07

Angular Momentum 08



Where k = radius of gyration

This is an expression for the kinetic energy of a body rolling on a horizontal surface without sliding.

Kinetic Energy of a Ring Rolling on a Horizontal Surface Without Sliding (Slipping):

Angular Momentum 11

This is an expression for Kinetic energy of a ring rolling on a horizontal surface without sliding (slipping).

Kinetic Energy of a Disc Rolling on a Horizontal Surface Without Sliding (Slipping):

Angular Momentum 10

This is an expression for Kinetic energy of a disc rolling on a horizontal surface without sliding (slipping).



Kinetic Energy of a Solid Sphere Rolling on a Horizontal Surface Without Sliding (Slipping):

Angular Momentum 12

This is an expression for Kinetic energy of a solid sphere rolling on a horizontal surface without sliding (slipping).



Kinetic Energy of a Hollow Sphere Rolling on a Horizontal Surface Without Sliding (Slipping):

Angular Momentum 13

This is an expression for Kinetic energy of a hollow sphere rolling on a horizontal surface without sliding (slipping).



Angular Momentum 14

Velocity and Acceleration of Body Rolling on Inclined Plane:

Velocity of a Body Rolling on an Inclined Plane Without Sliding (Slipping):

  • Consider a rigid body of mass ‘M’ and radius ‘R’ rolling down an inclined plane of inclination ‘θ’ from height ‘h’. 2. In this case, at the highest point body has potential energy and no kinetic energy. As the body rolls down the plane its potential energy decreases and gets converted into kinetic energy.

Angular Momentum 16This is an expression for the speed of body rolling down an inclined plane.

Velocities of Bodies Rolling on an Inclined Plane Without Sliding (Slipping):

Angular Momentum 17

Expression for an Acceleration of a Body Rolling on an Inclined Plane Without Sliding (Slipping):

  • The body is rolling down the inclined plane. Its initial velocity u = 0. Let ‘a’ be the acceleration of the body.

Angular Momentum 18

This is an expression for acceleration of body rolling down an inclined plane.



Acceleration of Bodies Rolling on an Inclined Plane Without Sliding (Slipping):

Angular Momentum 19



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