# 20.6 Conservation of angular momentum (application)  (Page 5/6)

 Page 5 / 6

## Conservation of angular momentum about two parallel axes

Example 4

Problem : Two uniform disks of masses “4M” and “M” and radius “2R” and “R” respectively are connected with a mass-less rod as shown in the figure. Initially, the smaller disk is rotating clockwise with angular velocity “ω” with the help of an electric motor mounted on it, whereas the larger disk is stationary. At an instant, the smaller disk reverses the direction of rotation, keeping its angular speed same. Find the angular velocity of the larger disk. The question involves two axes of rotation : one for larger plus smaller disks (z-axis) and other for smaller disk (z’-axis).

Solution : This question involves two axes of rotation : one for "larger plus smaller disks" (z-axis) and other for smaller disk (z’-axis). The treatment of angular momentum, in this case, differs from other cases in one important aspect. The constituents of the system are defined with respect to axis of rotation – not with respect to objects.

The body about the z - axis is a composite body comprising of larger and smaller disks. The smaller disk is also part of this composite body. The smaller disk (part of the composite body) rotates about its own axis (z’) parallel to z-axis. In the nutshell, the system comprises of two rotating bodies :

1. Composite body comprising of larger and smaller disk rotating about z-axis
2. Smaller disk rotating about z’-axis

We should note that the MI of the composite body is constant and is independent of the motion of smaller disk, because mass distribution about z-axis does not change by the rotation of smaller disk. It is so, because disks are uniform and circular in shape.

Now, let us check external torques on the system. The weight of smaller disk constitutes a torque, but it is perpendicular to the axis of rotation. Further, the torque due to smaller disk is balanced by an equal an opposite torque applied on the axle by the ground. This is confirmed, as there is no rotation of smaller disk in vertical plane. As such, we can conclude that there is no torque on the system in vertical direction and, therefore, we can apply law of conservation of momentum about z and z’ axes. Since two axes point in the same direction, the net angular momentums "before" and "after" are simply the arithmetic sum of angular momentums about the two axes. Let “C” and “S” subscripts denote the composite body and smaller disk respectively, then :

$\begin{array}{l}{L}_{i}={L}_{f}\\ {L}_{\mathrm{Ci}}+{L}_{\mathrm{Si}}={L}_{\mathrm{Cf}}+{L}_{\mathrm{Sf}}\end{array}$

The moment of inertia of the smaller disk about z’-axis is :

$\begin{array}{l}{I}_{S}=\frac{M{R}^{2}}{2}\end{array}$

Now, the moment of inertia, “ ${I}_{C}$ ”, of the composite body is given by the theorem of parallel axes as :

$\begin{array}{l}{I}_{C}=\frac{4Mx{\left(2R\right)}^{2}}{2}+\frac{M{R}^{2}}{2}+M{\left(2R\right)}^{2}=12.5M{R}^{2}\end{array}$

Let “ ${\omega }_{f}$ ” be the final angular velocity of the composite body. Substituting in the equation of conservation of angular momentum,

$\begin{array}{l}⇒{I}_{C}\phantom{\rule{2pt}{0ex}}x\phantom{\rule{2pt}{0ex}}0-{I}_{S}\phantom{\rule{2pt}{0ex}}x\phantom{\rule{2pt}{0ex}}\omega ={I}_{C}\phantom{\rule{2pt}{0ex}}x\phantom{\rule{2pt}{0ex}}{\omega }_{f}+{I}_{S}\phantom{\rule{2pt}{0ex}}x\phantom{\rule{2pt}{0ex}}\omega \end{array}$

Negative sign indicates that smaller disk rotates in clockwise direction.

$\begin{array}{l}⇒{I}_{C}\phantom{\rule{2pt}{0ex}}x\phantom{\rule{2pt}{0ex}}{\omega }_{f}=2{I}_{S}\omega \end{array}$

$\begin{array}{l}⇒{\omega }_{f}=\frac{2{I}_{S}\phantom{\rule{2pt}{0ex}}x\phantom{\rule{2pt}{0ex}}\omega }{{I}_{C}}=\frac{2\left(\frac{M{R}^{2}}{2}\right)\omega }{12.5M{R}^{2}}=\frac{\omega }{12.5}\end{array}$

## System consisting of rotational and translational motion

Example 5

#### Questions & Answers

What are the system of units
Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
58asagravitasnal firce
Amar
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad
what is the definition of resolution of forces
Atinuke Reply
what is energy?
James Reply
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
أوك
عباس
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply

### Read also:

#### Get the best Physics for k-12 course in your pocket!

Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics for k-12' conversation and receive update notifications?   By By Eric Crawford   By Edward Biton   