Angular Momentum

Angular Momentum 

At Higher Level, the concept of Linear Momentum ( See Collisions and Impulse ) was discussed. As we have seen in previous sections, for all Linear Dynamics quantities there is an equivalent Angular Dynamics quantity. 

The Angular Momentum ( L ) of an object is defined as the Moment of the Linear Momentum. 

The video below shows an explanation of Angular Momentum and its applications by Astronaut Mike Fossum on the International Space Station.

The Diagram below shows an object of mass m rotating around a central point P at a distance of r. 

 The Object has a Linear Momentum equal to :-

p  =  m v

The Moment of Momentum ( L ) is therefore equal to :-

L  =  m v r

And by substituting for angular velocity :- 

L  =  mr2ω 

Finally by substituting for Moment of Inertia :-

L  =  I ω

Angular Momentum ( L ) has units of kg m2 s-1.

Note - The above derivation applies for a orbiting Point Mass, hence why I =  mr2. This term will therefore change depending upon what type of object is rotating. However, the general form of 

L = Iω will always apply. At Advanced Higher Level, only the derivation for a Point Mass is examinable.

Conservation of Angular Momentum

As was seen in the Higher Physics Course, in the absence of external Forces, Momentum is conserved and the same applies for Angular Momentum :-

" The Total Angular Momentum before a collision will equal the Total Angular Momentum after a collision, as long as no external Torques act upon the system. "

The video below shows a practical application of Conservation of Angular Momentum. 

Note - In the above video, Principal Skinner and the container have initially no Angular Momentum. Principal Skinner begins to run clockwise and in order to conserve Angular Momentum for the system, the container must begin to rotate anti-clockwise. 

Example 1 -

Two Balls are moving in a circular path of radius 1m as shown above. The Green Ball has a mass of 1kg and moves clockwise. The red ball has a mass of 2 kg and moves anti-clockwise. The two balls are both travelling at 12 ms-1 then collide and stick together.

What is the magnitude and direction of the Velocity of the balls after the collision ? 

Total Angular Momentum before  =  Total Angular Momentum after

( m1 V1 r  ) + ( m2 V2 r )  =  ( m1 + m2 ) V3 r

( 1 x 12 x 1 ) + ( 2 x -12 x 1 )  =  ( 3 x V3 x 1 ) 

12 + ( -24 )  =  3V3

V3  =  - 4

V  =  4 ms-1 Anti-Clockwise 

Example 2 -

All Comets travel in elliptical orbital paths, with the Sun at one of the foci. Explain, in terms of Angular Momentum, why the Velocity of a Comet increases in magnitude as it approaches the Sun.

As the Comet approaches the Sun, the radius of its orbit decreases. Due to this, the Comet's Moment of Inertia also decreases. In order to conserve the Comet's Angular Momentum, the Velocity of the Comet must increase. 

Example 3 - Exam Style Question

A turntable rotating at a rate of 15 revolutions per minute has a mass of 60 grams dropped vertically onto it at a point 0.12 m from the axis of rotation. The System as a whole continues to rotate at a rate of 10 revolutions per minute after the collision. 

Use the Principle of Conservation of Angular Momentum to calculate the Moment of Inertia for the Turntable.  

Moment of Inertia of the Turntable  = Y 

Moment of Inertia of the 60 g mass  =  mr2

Initial Angular Velocity  ( ω0 )  =   15 rev min-1  =  ( 15 x 2π ) / 60  =  1.57 Rad s-1

Final Angular Velocity ( ω )  =  10 rev min-1  =  ( 10 x 2π ) / 60  =  1.05 Rad s-1

Total Angular Momentum Before  =  Total Angular Momentum After

0  =  ( Iturntable + Imass ) x ω

Y x 1.57  =  ( Y + 8.64x10-4 ) x 1.05

Y  =  9.072x10-4 / 0.52  =  1.74x10-3  kgm2

Rotational Kinetic Energy 

As can be seen above, the Moment of Inertia is the rotational analogue of linear Mass in calculations. This can be shown to apply for Kinetic Energy calculations. If an Object with Moment of Inertia of I rotates around a fixed point at a constant Angular Velocity of ω rad s-1, then its Rotational Kinetic Energy can be found by :-

Ek  =  1/2 I ω2

Where :-

 Ek  =  Rotational Kinetic Energy  ( J ) 

I  =  Moment of Inertia ( kg m2

ω  =  Angular Velocity ( rad s-1

Note - An object rolling down a slope will have Two Distinct Kinetic Energy Types - Rotational Kinetic Energy as it rolls, and Linear Kinetic Energy as it moves down the slope. 

Rotational Work Done 

In Linear Mechanics, Work Done can be found by Multiplying the Force by the Distance that Force acts over ( Ew  =  F x d ) . The Rotational Work Done can be found by the using the Rotational Equivalents :-

Ew  =  Tθ

Where :-

Ew  =  Work Done ( J ) 

T  =  Torque Applied ( Nm ) 

θ  =  Angular Displacement ( rad ) 

By applying a Torque to an Object its Rotational Kinetic Energy will change following :-

Ew  = Δ Ek 

Tθ  =  ( 1/2I ω2 )  - (  1/2I ω02

where :-

T  =  Torque Applied ( Nm ) 

θ  =  Angular Displacement ( rad ) 

I  =  Moment of Inertia ( kg m2

ω0  = Initial Angular Velocity ( rad s-1 )

ω  =  Final Angular Velocity ( rad s-1 )