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Solid Objects

Up to now we have only looked at point like masses, where effectively all the mass is concentrated at a single location. However things get a little more complicated with ‘rigid bodies’. This is where the mass of an object is distributed across a wide enough area that it affects the rotational motion of the object. The usual example that is given is of an ice skater whose rotation rate speeds up as they draw their arms in. Another way of thinking about this is to take a wooden rod such as a broom handle and try to turn it so that it completes one rotation per second. The ease with which you can do this, and the amount of energy stored, depends of the direction and position of the axis of rotation even though the amount of mass and the angular velocity is constant! We have to take account of how the mass is distributed about the axis of rotation.

 Rotating Rod

To deal with an extended object it looks as if we have to deal with every little particle of mass separately like this:

 Moment of Inertia of an Irregular Shape

Here we have an irregularly shaped, rigid object. For convenience it is a plane but it could also be a three dimensional object, the diagram would however be a little more complex. The object is made up of a huge number of small particles, one of which has been shown. The particle has a mass m and is at a distance r from the axis of rotation. The kinetic energy of the particle is given by:

 Equation.

To deal with an extended object it looks as if we have to deal with every little particle of mass separately like this:

 Equation.

But the object is rigid and so the angular velocities of all the particles are the same. This means that the kinetic energy becomes:

 Equation.

You can think of the expression:

 Equation.

as being the rotational equivalent of inertial mass; a measure of how hard or easy it is to start an object rotating or to stop it later and of how much energy is stored in it when it is rotating. In fact we give this quantity a special name, the moment of inertia and give it the symbol, I. If we are dealing with a continuous object such as a disc or a bar rather than a group of individual masses then we can replace the summation with an integration:

Rather than spending several screens deriving the moments of inertia of different shapes the most important equations are presented here:

Object

Moment of Inertia

1

 Thin Rod: Centre
Thin rod or bar, rotating about the centre

Equation.

2

 Thin Rod: End
Thin rod or bar, rotating about one end

Equation.

3

 Rectangular Plate: Centre
Rectangular plate rotating about the centre

Equation.

4

 Thin Plate: Edge
Thin rectangular plate rotating about one edge

Equation.

5

 Solid Cylinder
Solid cylinder rotating about the central axis

Equation.

6

 Thick, Hollow Cylinder: Centre
Thick walled cylinder rotating about the central axis

Equation.

7

 Thin Hollow Cylinder, Centre
Thin walled cylinder rotating about the central axis

Equation.

8

 Solid Sphere: Centre
Solid sphere rotating about the central axis

Equation.

9

 Hookes Law
Thin walled hollow sphere rotating about the central axis

Equation.

10

 Parallel Axis Theorum.
Thick walled cylinder rotating about the central axis
A general body with moment of inertia Icm through its centre of mass, rotating about a parallel axis, a distance d from the centre of mass

Equation.
(Parallel Axis Theorem)

The parallel axis theorem tells us that the moment of inertia of an object rotating about an axis that is distant from its centre of mass is made of two distinct components. The first, Icm, is just the moment of inertia of the object rotating about a parallel axis that passes through the centre of mass. The second, Md2, is just the moment of inertia of a point mass rotating about the axis at a distance d.

A student looks at the equation for a thick walled, hollow cylinder rotating about its central axis and extends the idea to a thick walled, hollow sphere. They suggest that the moment of inertia of the shape is:

 Equation.

By analysing the extreme cases of the equation say whether you believe the equation could be correct.

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