# Unwinding Hoop: what holds it up?

1. Jun 19, 2015

### omega5

1. The problem statement, all variables and given/known data
As shown in the figure, a string is wrapped several times around the rim of a small hoop with radius R and mass M. The free end of the string is pulled upward in just the right way so that the hoop does not move vertically as the string unwinds.

2. Relevant equations
$\tau=I\alpha$
$\tau=FR$
$I=MR^2$
$\Sigma F = ma$

3. The attempt at a solution
Assumptions:
The tangential acceleration of the string is equal to the tangential acceleration of the hoop, which is essentially rolling along the string. This acceleration is due to a net force given by $F_h-F_T$, where $F_h$ is the force exerted by hand and $F_T$ is the tension force.
The torque (and angular acceleration) will be greater than zero and negative (clockwise) since $F_h>F_T$.
The net force acting on the center of mass is zero.

What force acts at the center of mass to keep the hoop in equilibrium? Getting stuck on this part has prevented me from understanding the situation enough to solve for tension, angular acceleration or the force exerted by the hand.

2. Jun 19, 2015

### paisiello2

Is the hoop in equilibrium? I think it is rotating. But what point is it rotating about?

3. Jun 19, 2015

### Dick

The total effect of gravitation is equivalent to a force acting at the center of mass. The tension is not acting at the center of mass. But if their sum is zero then there is no vertical acceleration regardless of 'where' each acts. So you should get started with the other parts of the exercise.

4. Jun 19, 2015

### omega5

Thanks for responding! I should have been clearer, I meant equilibrium along the y-axis.

5. Jun 19, 2015

### omega5

Thank you for the clear explanation. One more, nitpicky question: how does the force provide upward acceleration and not just torque? I just want to understand everything that's going on.

6. Jun 20, 2015

### haruspex

An individual force is not usually thought of as providing acceleration. It's the net of all the forces that provides the acceleration. If the tension equals the weight there is no net vertical force.

7. Jun 20, 2015

### omega5

I'm glad you pointed that out. Come to think of it, that has contributed to my mistakes on other problems.
So, at the point just before the string leaves the hoop, the tension is acting upward on the hoop but downward on the point of string there?

8. Jun 20, 2015

### haruspex

Yes, the hoop pulls down on the string just as the string pulls up on the hoop... action and reaction.

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