Angular momentum thin string problem

In summary, we have a thin string wrapped around a cylindrical hoop of radius R and mass M. As the string unwinds, the hoop falls vertically from rest with one end of the string fixed. The angular momentum of the hoop about its center of mass can be determined as a function of time, and the tension in the string also varies as a function of time. The tension has an x-component that causes the hoop to be pushed sideways until the point where the string leaves the hoop is directly underneath the point where the string is held. This can be calculated using the moment of the tension around the center of mass.
  • #1
aquafina
1
0
A thin string is wrapped around a cylindrical hoop of radius R and mass M. One end of the string is fixed and the hoop is allowed to fall vertically starting from rest, as the string unwinds. a)Determine the angular momentum of the hoop about its CM as a function of time. b)What is the tension in the string as a function of time.

I'm having a hard time even picturing this and how the hoop would rotate as the string unwinded. Some hints would be really helpful, thanks.
 
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  • #2
Does this help a bit?
 
Last edited:
  • #3
aquafina said:
A thin string is wrapped around a cylindrical hoop of radius R and mass M. One end of the string is fixed and the hoop is allowed to fall vertically starting from rest, as the string unwinds. a)Determine the angular momentum of the hoop about its CM as a function of time. b)What is the tension in the string as a function of time.
I'm having a hard time even picturing this and how the hoop would rotate as the string unwinded. Some hints would be really helpful, thanks.


Im not sure about A (just starting this in my class), but it seems logical that the object is in free fall, so would their even be a tension?. But then again, it would take a force to make the string unwind..:confused:
 
  • #4
If you investigate my suggestion you will find that the tension has an x-component which cannot be canceled by any other force. This means that the hoop moves until the tension has no x-component! The hoop is therefore pushed sideways by the tension until the point where the string leaves the hoop is underneath the point where the string is held.Taking the moment of the tension around the com gives:
[tex]\begin{align*}
TR=I\alpha\\
=I\frac{d\omega}{dt}\\
TRdt=Id\omega\\
TR\int_{0}^{t}dt=I\int_{0}^{\omega}d\omega\\
...
\end{align*}[/tex]
 
Last edited:

What is the "Angular Momentum Thin String Problem"?

The "Angular Momentum Thin String Problem" is a physics problem that involves a thin, massless string wrapped around a pulley, with a weight attached to one end of the string. The string and weight system can rotate around the pulley, and the problem involves calculating the angular momentum of the system at different points in time.

What are the key concepts involved in solving the "Angular Momentum Thin String Problem"?

The key concepts involved in solving the "Angular Momentum Thin String Problem" include angular momentum, rotational motion, and conservation of angular momentum. It is also important to understand the properties of a thin, massless string and the forces acting on the system, such as tension and friction.

What are the steps to solve the "Angular Momentum Thin String Problem"?

To solve the "Angular Momentum Thin String Problem," you should first draw a diagram of the system and label all the known variables. Then, use the formula for angular momentum (L = Iω) to calculate the angular momentum at different points in time. Next, apply the principle of conservation of angular momentum to set up an equation and solve for any unknown variables. Finally, check your answer and make sure it is reasonable and consistent with the given information.

What are some common mistakes when solving the "Angular Momentum Thin String Problem"?

Some common mistakes when solving the "Angular Momentum Thin String Problem" include forgetting to account for all the forces acting on the system, using the wrong formula for angular momentum, and not considering the direction of the angular momentum. It is also important to be careful with units and make sure they are consistent throughout the calculation.

How is the "Angular Momentum Thin String Problem" relevant in real-world situations?

The "Angular Momentum Thin String Problem" is relevant in real-world situations, such as in the design and operation of pulley systems, which are used in many machines and devices. Understanding the principles of angular momentum and rotational motion is also important in fields such as mechanics, engineering, and astronomy.

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