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if the mass of a wheel is increased by a factor of 2 and the radius is increased by a factor of 1.5, by what factor is the moment of inertia increased? model the wheel as a solid disc.

Answer :

The Moment of inertia is increased by the factor of 3.

  1. Rotational inertia, or moment of inertia, is the rotational equivalent of mass; this is the quantity we want to measure to see how much energy is needed to accelerate the wheel.
  2. We can measure the weight of a wheel relatively easily, but it is not necessarily true that the lightest wheel will have the smallest moment of inertia or vice versa. If we have a rigid body rotating with an angular velocity Ω about a fixed axis , each particle in it will have a certain amount of kinetic energy.
  3. A particle of mass m with radius r from the axis of rotation moves in a circle with radius r with angular velocity Ω around this axis. So it has a linear velocity v = Ωr and its kinetic energy K is ½ mv2 = ½ mΩ2r2.
  4. The total kinetic energy is the sum of the kinetic energies of its particles, and each particle in a solid moves with the same angular velocity Ω (it moves the same number of angular degrees per unit time), but the radius r can be different for different particles. So the total kinetic energy is

K = ½ (m1r12 + m2r22 + …)Ω2 = ½ (∑miri2)Ω2

The moment of inertia of the wheel is given as

I = MR2

if M' = 2M and R' = 1.5R

then I' = M'R'2 = (2M)(1.5R)2 = 3.0 MR2 = 3I

so the factor is 3

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