Rotation of a spool about rough ground

In summary, the problem involves a spool of thread with a tightly wound thread and two caps of different radii. The spool is placed on the ground and the thread is pulled causing the spool to roll without slipping. The magnitude of the friction force is equal to (\frac{I + mR_1R_2}{I + mR_2^2}T), where T is the tension of the thread and I is the moment of inertia of the spool. To solve this problem, Newton's second law and torque equations are applied, with the main issue being a sign convention for the torque equation. By adjusting the signs, the correct answer is found to be f = (\frac{I + mR_
  • #1
bigevil
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Homework Statement



A spool of thread comprises a cylinder of radius R1 and is capped with two disks of radius R2, where R2 > R1. Some thread is tightly wound over the cylinder. The whole spool is laid stationary, side down on the ground and the thread is pulled. The spool rolls without slipping. Show that the magnitude of friction force is

[tex]f = (\frac{I + mR_1R_2}{I + mR_2^2}T)[/tex], where the tension of the thread pulled is T and the moment of inertia of the spool is I.

The Attempt at a Solution



This is not a really difficult problem. I don't really have a big problem with the general method, but I do have some problems arriving at the exact final answer.

Applying Newton's second law,

[tex]T - f = ma_{cm}[/tex] for the centre of mass of the spool.

Applying torques about the centre of the spool

[tex]TR_1 - fR_2 = I\alpha[/tex]

Also, given pure rolling, [tex]\alpha = \frac{a_c}{R_2}[/tex]. Substituting this, I get roughly the same expression [tex]f = (\frac{I - mR_1R_2}{I - mR_2^2}T)[/tex].

Assuming I got the general steps right, the main thing I got wrong here is the torque equation. Had I reversed the signs I would have gotten [tex]-TR_1 + fR_2 = I\alpha[/tex], and the correct answer, but I can't understand how this is so. Can someone help me to see the light??
 
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  • #2
It's a sign convention issue. To apply the condition for rolling without slipping, the rotational and translational accelerations must have the same sign. Since the torque due to friction acts to increase the rotational acceleration, it should have the same sign in your torque equation.
 
  • #3
Thanks Al.
 

1. What is the concept of rotation of a spool about rough ground?

The rotation of a spool about rough ground refers to the motion of a spool as it rolls along a rough surface, causing it to rotate in a circular motion.

2. How does the roughness of the ground affect the rotation of a spool?

The roughness of the ground can impact the rotation of a spool by creating friction between the spool and the surface. This friction can cause the spool to roll slower or faster, and can also affect the direction of its rotation.

3. What is the relationship between the diameter of the spool and its rotation on rough ground?

The diameter of the spool can affect its rotation on rough ground by changing the amount of surface area that is in contact with the ground. A larger diameter spool may experience less friction and roll more smoothly, while a smaller diameter spool may experience more friction and roll slower.

4. Can the shape of the spool impact its rotation on rough ground?

Yes, the shape of the spool can affect its rotation on rough ground. A circular spool may roll more smoothly than a non-circular spool, which may have uneven contact with the ground and experience more friction.

5. What other factors can affect the rotation of a spool on rough ground?

Other factors that can impact the rotation of a spool on rough ground include the weight of the spool, the surface material of the ground, and the speed at which the spool is rolling. These factors can all influence the amount of friction and the resulting rotation of the spool.

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