A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.7 Torque

Figure 79: A rotating bicycle wheel.

the tangential motion of the wheel, yields

M ̇v = f sin θ, (8.52)

where M is the mass of the wheel (which is assumed to be concentrated in the
wheel’s rim).

Let us now convert the above expression into a rotational equation of motion.

If ω is the instantaneous angular velocity of the wheel, then the relation between

ω and v is simply

v = b ω. (8.53)

Since the wheel is basically a ring of radius b, rotating about a perpendicular
symmetric axis, its moment of inertia is

I = M b^2. (8.54)

Combining the previous three equations, we obtain

where

I ω ̇ = τ, (8.55)

τ = f b sin θ. (8.56)

Equation (8.55) is the angular equation of motion of the wheel. It relates the

wheel’s angular velocity, ω, and moment of inertia, I, to a quantity, τ, which is

known as the torque. Clearly, if I is analogous to mass, and ω is analogous to

f sin f



O r^

P

b