How Do You Calculate Force and Work for a Suspended Spool in Rotational Motion?

[Minihoudini]

Homework Statement

A narrow but solid spool of thread has radius R and mass M. If you pull up on the thread so that the CM of the spool remains suspended in the air at the same place as it unwinds,
(a) what force must you exert on the thread?
(B) how much work have you done by the time the spool turns with angular velocity w?

Homework Equations

k=1/2Iw^2 + 1/2MR^2w^2

The Attempt at a Solution

I've tried to make an understanding that when your pulling upward for problem (a) you are canceling out the force being put on the spool. by pulling up. I'm just having trouble trying to find the correct formula to use to find Force. any help in the right direction would be great.

 

[kuruman]

Draw a free body diagram of the spool. How many forces act on the spool? How must these forces be related if the spool remains suspended in air, i.e. its center of mass does not accelerate?

 

[kerimek]

I would approach this problem by first considering the forces acting on the spool. When you pull up on the thread, you are applying a force in the upward direction. This force must be equal to the weight of the spool, which is given by the formula F = mg, where m is the mass of the spool and g is the acceleration due to gravity. In this case, the mass is M and the acceleration due to gravity is 9.8 m/s^2.

In order to keep the CM of the spool suspended in the air, the force you exert on the thread must also counteract the torque produced by the weight of the spool. The torque is given by the formula τ = Fr, where F is the force applied and r is the radius of the spool. Therefore, the force you must exert on the thread can be found by rearranging this formula to F = τ/r.

For part (b), the work done can be calculated by using the formula W = τθ, where τ is the torque and θ is the angular displacement of the spool. In this case, the torque is again given by τ = Fr, and the angular displacement can be calculated using the formula θ = ωt, where ω is the angular velocity and t is the time. Therefore, the work done can be calculated by the formula W = Frωt.

Substituting the values given in the problem, we get W = (mg)rωt. We can also use the formula ω = v/r, where v is the linear velocity of the spool. Therefore, we can also write the work done as W = (mg)vωt/r.

I hope this helps guide you in the right direction. Remember to always consider the forces and torques acting on the object when solving rotational motion problems.