Question

Fly Wheel: The fly wheel has a mass M , a radius R , and a moment of inertia I=1/2MR^2          . The fly wheel has a gear of radius Rg < R that is used to turn the wheel.

Drive Chain: The drive chain connects the gear on the fly wheel to a motor. The gear, the black circle in the middle, rotates with the wheel and the chain is pulling on the gear without slipping in the direction shown. You can neglect the mass of the chain. The tension in the top portion of the chain, caused by the motor, is Ft while the tension in the bottom portion of the chain is zero.
The goal of this system is to give the flywheel the greatest angular velocity in the shortest amount of time. However, there is a maximum force (Fm) that you can apply to the drive wheel and a maximum speed (vm) that you can pull the chain.
A) What is the relation between the angular velocity of the fly wheel (wv), and the velocity of the chain (vch) ? What is the relation between the angular acceleration of the fly wheel (aw) , and the acceleration of the chain (ach)    ?

B) M and R are constants but you can change the radius of the gear, (Rg) . If you want the fly wheel to move as fast as possible, do you want Rg to be large or small? Explain.

C) Write down the torque equation for the wheel and solve for the angular acceleration of the wheel, (aw) , in terms of the tension Ft and the other parameters in the problem.

Answer 1

part A:

angular velocity of fly wheel=angular velocity of the chain

==>Wv=Vc/Rg

==>Vc=Rg*Wv..(1)

angular acceleration of the wheel=angular acceleration of the chain

==>alpha_W=aC/Rg

==>aC=alpha_W*Rg

part B:

as torque=moment of inertia*angular acceleration

moment of inertia of the wheel is constant

so angular acceleration is directly proportional to torque

torque=force*radial distance

=Ft*Rg

so higher the Rg, higher the torque and higher the acceleration of the wheel

part C:

torque=Ft*Rg

moment of inertia of the wheel=0.5*M*R^2

so angular acceleration=torque/moment of inertia

=2*Ft*Rg/(M*R^2)