# Drive train problem

Hi all,
I was wondering if someone could help me out on this (taken from Newtonian Mechanics, A.P. French, pg 705, problem14-23)

A wheel of uniform thickness , of mass 10kg and radius 10cm is driven by a motor through a belt. The drive wheel on the motor is 2cm in radius. The motor is capable of delivering a torque of 5 Nm.

A) Assuming that the belt does not slip on the wheel, how long does it take to accelerate the large wheel from rest up to 100 rpm?

B) If the coefficient of friction between belt and wheel is 0.3, what are the tensions in the belt on the two sides of the wheel?
(Assume that the belt touches the wheel over half its circumference.)

For A, I had that if the belt does not slip, all of the torque is transferred to the driven wheel and I applied torque = moment X angular acceleration. And alpha = omega X time assuming alpha is constant.

For B, I thought of deriving friction from the normal force exerted by circumference on belt, but this looks like a nightmare to compute since the force the belt exerts on the circumference varies over the circumference.

In any case, does not the tension In a massless rope have to the the same on both sides always? If not we get infinite acceleration in any given differential element!

thanks.

## Answers and Replies

ee7klt said:
In any case, does not the tension In a massless rope have to the the same on both sides always? If not we get infinite acceleration in any given differential element!

thanks.

Welcome to PF ee7klt!

For a rotating massless pulley, $$\alpha \neq 0$$ but moment of inertia is zero as mass is zero. Hence, the tensions have to be same on either side of the pulley (you can see why this is so by making a freebody diagram of a pulley whose center of mass is attached to a rope the tension in which is T and which is wrapped by an inextensible rope--now find the tensions in either part of the rope using the idea just mentioned).

If however, friction exists, the analysis changes a bit (you have to consider an element of the rope), you have to use a relation involving the (different) tensions, T1 and T2 on either side of the pulley, the coefficient of static friction of the pulley-rope interface and the angle of wrap. Think about this.

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