Measurement of the moment of inertia of a flywheel

A flywheel of radius R is set up on a horizontal axle of radius r. A string of length h is wrapped round the axle with a mass m tied to the end (Figures 1 and 2). The moment of inertia of the flywheel and axle is I. The flywheel is accelerated by the couple applied by the mass m. The mass is allowed to fall through a height h at which point the string leaves the axle. The velocity of the falling mass at this instant is v and the angular velocity of the flywheel ω.

The potential energy lost by the weight is converted into kinetic energy of the weight, kinetic energy of the flywheel and heat due to friction in the bearings.

If the energy lost per revolution due to friction is E and the flywheel makes n₁ revolutions during acceleration, then:

mgh = ½ mv² + ½ Iω² + n₁E

The flywheel is then allowed to come to rest due to the frictional couple. If it stops after a further n₂ revolutions then:

½ Iω² = n₂E

Therefore:

We could convert linear velocity (v) into angular velocity (ω) if we wished using v = Rω.

Now the angular velocity ω at the end of the period of the acceleration is given by:

ω/2 = 2πn₁/t

Since ω>/2 is the average angular velocity of the flywheel and 2πn₁ is the angular distance covered by any point on it in a time t.

Hence the moment of inertia of the flywheel can be calculated.

Measurement of the moment of inertia of a disc

The conservation of angular momentum may be used to measure the moment of inertia of a disc.

If a disc of moment of inertia I₁ is dropped coaxially on to a freely rotating table of known moment of inertia I_o and angular velocity w_o (Figure 3) then:

momentum of system before disc is dropped = momentum of system after disc is dropped

I_oω_o = (I_o + I₁)ω₁

where ω₁ is the angular velocity after the disc has been dropped onto the rotating table.

This is thus a very simple method of determining I₁, the moment of inertia of the disc.

If a disc is dropped onto a table that is maintained at a constant angular velocity (such as a record deck) then a frictional couple must act to accelerate the disc. The energy while the disc is accelerating goes both to accelerate the disc and to maintain the constant velocity of the table.