Unwinding Hoop: what holds it up?

[omega5]

Homework Statement

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.

Homework Equations

##\tau=I\alpha##
##\tau=FR##
##I=MR^2##
##\Sigma F = ma##

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.

 

[paisiello2]

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

 

[Dick]

Homework Statement

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.

Homework Equations

##\tau=I\alpha##
##\tau=FR##
##I=MR^2##
##\Sigma F = ma##

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.

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.

 

[omega5]

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

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

 

[omega5]

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.

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.

 

[haruspex]

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.

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.

 

[omega5]

An individual force is not usually thought of as providing acceleration. It's the net of all the forces that provides the acceleration.

I'm glad you pointed that out. Come to think of it, that has contributed to my mistakes on other problems.

If the tension equals the weight there is no net vertical force.

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?

 

[haruspex]

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?

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