# Unwinding Hoop: what holds it up?

• omega5
In summary: So at the point of contact, the tension is acting downward on the string. In summary, the string is wrapped around a small hoop with radius R and mass M, and the free end is pulled upward to provide an upward acceleration. The tangential acceleration of the string is equal to the tangential acceleration of the hoop, and the net force acting on the center of mass is zero. The force of tension and the force of gravity are equal and opposite, providing a net torque of zero and allowing the hoop to remain in equilibrium. At the point of contact, the tension is acting downward on the string and upward on the hoop, creating a balanced force.
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.

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

omega5 said:

## 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
paisiello2 said:
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.

Dick said:
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.

omega5 said:
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
haruspex said:
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.
haruspex said:
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?

omega5 said:
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.

omega5

## 1. What is an unwinding hoop?

An unwinding hoop is a scientific apparatus used to demonstrate the concept of centrifugal force. It consists of a metal hoop with a string attached, and when spun, the string is pulled outwards due to the centrifugal force, causing the hoop to remain upright.

## 2. What holds up the unwinding hoop?

The principle of inertia is what holds up the unwinding hoop. The hoop's spinning motion creates a centrifugal force that opposes the inward pull of gravity, keeping the hoop upright.

## 3. How does the unwinding hoop illustrate centrifugal force?

The unwinding hoop demonstrates centrifugal force by showing how a rotating object can create a force that pulls objects away from the center of rotation. In this case, the hoop's rotation creates a centrifugal force that pulls the string outwards, keeping the hoop upright.

## 4. What factors affect the stability of the unwinding hoop?

The stability of the unwinding hoop is affected by the hoop's shape, size, and material, as well as the speed and direction of rotation. A larger hoop with a wider diameter and a higher rotational speed will have a greater centrifugal force, making it more stable.

## 5. Can the unwinding hoop be used for more than just a demonstration?

Yes, the unwinding hoop can be used for various applications, such as measuring rotational speed or acceleration. It can also be used in experiments to study the effects of different forces on rotating objects. Additionally, it can be used as a toy or a form of entertainment.

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