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25.

Problem 12.58E (HRW)

With centre and spokes of negligible mass, a certain bicycle wheel has a thin rim of radius and weight ; it can turn on its axle with negligible friction. A man holds the wheel above his head with the axle vertical while he stands on a turntable free to rotate without friction; the wheel rotates clockwise, as seen from above, with an angular speed of , and the turntable is initially at rest. The rotational inertia of wheel + man+ turntable about the common axis of rotation is . The man’s free hand suddenly stops the rotation of the wheel (relative to the turntable). Determine the angular velocity (magnitude and direction of the system).

Solution:             Click For PDF Version

In this problem data has been given in the British fps system. In this system of units the basic unit of mass is slug, of length is ft, and of force is lb. Slug and lb are connected by the relation that a mass of 1 slug under acceleration of 32 ft s-2 experiences a force of 1 lb.

There is no external torque on the system about the direction of the vertical axis. Therefore, angular momentum is a constant of motion.

Radius of the bicycle wheel .

Weight of the wheel =8.36 lb.

Therefore, mass of the wheel =

= 0.259 slug.

Moment of inertia of the wheel, I =

=

=0.337 slug ft2

Wheel is spinning with speed, .

Angular momentum of the wheel, which is also the angular momentum of wheel+ man+ turntable,

.

Rotational inertia of wheel+ man+ turntable

As no external torque is exerted on the system when the rotation of the wheel is stopped by the man using his free- hand (relative to turntable), the angular momentum of the system will be L. The rotational angular speed of the system can now be calculated using the definition

The direction of rotation will continue to remain clockwise when seen from top.