Moment of Inertia and Rotational Kinetic Energy


Quick
The greater the moment of inertia, the greater the kinetic energy of a rigid body rotating with a given angular speed. The greater a body's moment of inertia, the harder it is to start the body rotating if it's at rest and the harder to stop its rotation if it's already rotating. For this reason, the moment of inertia is also called the rotational inertia.


Equations
(Eq1)    
I =
 
i
mi ri2
Moment of inertia
(Eq2)    
K =  
1
2
2
Rotational kinetic energy of a rigid body


Nomenclature
mmass
rperpendicular distance from axis
vspeed
ωangular speed


Details

The rapidly rotating blade of a table saw has kinetic energy due to that rotation. But can that energy be expressed. The familiar formula K = (1/2)mv2 cannot be applied to the saw as a whole because that would only give the kinetic energy of the saw's center of mass, which is zero.

Instead, the table saw is treated as a collection of particles with different speeds. The kinetic energies of all the particles can then be added up to find the kinetic energy of the body as a whole.

A rotating rigid body consists of mass in motion, so it has kinetic energy. This kinetic energy can be expressed in terms of the body's angular velocity and a quantity called moment of inertia. To develop this relatinship, think of the body as being made up of a large number of particles, with masses m1, m2, ... , at distances r1, r2, ... from the axis of rotation. The particles are labeled with the index i: The mass of the ith particle is mi, and its distance from the axis of rotation is ri. The particles don't necessarily all lie in the same plane, so we specify that ri is the perpendicular distance from the axis to the ith particle.

When a rigid body rotates about a fixed axis, the speed vi of the ith particle is given by eq? from the lesson?, vi = riω, where ω is the body's angular speed. Different particles have different values of r, but ω is the same for all (otherwise, the body wouldn't be rigid). The kinetic energy of the ith particle can be expressed as:

1
2
mivi2 =  
1
2
miri2ω2

The total kinetic energy of the body is the sum of the kinetic energies of all its particles:

K =  
1
2
m1r12ω2 +  
1
2
m2 r22ω2 + ⋅⋅⋅ =
 
i
1
2
mi ri2ω2

Taking the common factor ω2/2 out of this expression, the following is:

K =  
1
2
(m1r12 + m2 r22 + ⋅⋅⋅)ω2 =  
1
2
(
 
i
mi ri2 )ω2

The quantity in parentheses on the right side of the equation above, obtained by multiplying the mass of each particle by the square of its distance from the axis of rotation and adding these products, tells how the mass of the rotating body is distributed about its axis of rotation, is denoted by I, and is called the moment of inertia of the body for this rotation axis. This quantity is also called the rotational inertia.

I = m1r12 + m2 r22 + ⋅⋅⋅ =
 
i
mi ri2

or

(Eq1)    
I =
 
i
mi ri2

Eq1 is the definition of the moment of inertia. The word "moment" means that I depends on how the body's mass is distributed in space; it has nothing to do with a "moment" of time. For a body with a given rotation axis and a given total mass, the greater the distance from the axis to the particles that make up the body, the greater the moment of inertia. In a rigid body, the distances ri are all constant and I is independent of how the body is rotating around the given axis. The SI unit of moment of inertia is the kilogram-meter2 (kg ⋅ m2).

In terms of moment of inertia I, the rotational kinetic energy K of a rigid body is:

(Eq2)    
K =  
1
2
2

The kinetic energy given by Eq2 is not a new form of energy; it's the sum of the kinetic energies of the individual particles that make up the rigid body, written in a compact and convenient form in terms of the moment of inertia. When using Eq2, ω must be measured in radians per second, not revolutions or degrees per second, to give K in joules; this is because vi = riω was used in the derivation.

Eq2, which gives the kinetic energy of a rigid body in pure rotation, is the angular equivalent of the formula K = (1/2)Mv2, which gives the kinetic energy of a rigid body in pure translation. In both furmulas there is a factor of (1/2). Where mass M appears in one equation, I (which involves both mass and its distribution) appears in the other. Finally, each equation contains as a factor the square of a speed — translational or rotational as appropriate. The kinetic energies of translation and of rotation are not different kinds of energy. They are both kinetic energy, expressed in ways that are appropriate to the motion at hand.

Eq2 gives a simple physical interpretation of moment of inertia: the greater the moment of inertia, the greater the kinetic energy of a rigid body rotating with a given angular speed ω. The kinetic energy of a body equals the amount of work done to accelerate that body from rest. So the greater a body's moment of inertia, the harder it is to start the body rotating if it's at rest and the harder it is to stop its rotation if it's already rotating. For this reason, I is also called the rotational inertia.

One may be tempted to try to compute the moment of inertia of a body by assuming that all the mass is concentrated at the center of mass and then multiplying the total mass by the square of the distance from the center of mass to the axis. Resist that temptation; it doesn't work! For example, when a uniform thin rod of length L and mass M is pivoted about an axis through one end, perpendicular to the rod, the moment of inertia is I = ML2/3. If the mass was taken to be concentrated at the center, a distance L/2 from the axis, the incorrect result would be obtained: I = M(L/2)2 = ML2/4.



Related
▪ L - Second Moment, or Moment of Inertia, of an Area
▪ L - Polar Moment of Inertia
▪ L - Parallel Axis Theorem
▪ P - Moment of Inertia and Couples