Moment of Inertia Homework: Solving for Force P

[totalmajor]

Homework Statement

A light, flexible rope is wrapped several times around a hollow cylinder with a weight of 55.0 N and a radius of 0.25 m, that rotates without friction about a fixed horizontal axis. The cylinder is attached to the axle by spokes of a negligible moment of inertia. The cylinder is initially at rest. The free end of the rope is pulled with a constant force P for a distance of 3.00 m, at which point the end of the rope is moving at 4.00 m/s. If the rope does not slip on the cylinder, what is the value of P?

Homework Equations

v=$$\omega$$r
$$\alpha$$=$$\tau$$/I
I=MR^2

The Attempt at a Solution

Alright, I honestly don't know where to start with this problem. It's review we're doing and I know the basics of inertia but nothing past that!

Can anybody help me start?

Thanks alot,
Peter

 

[andrevdh]

work done by P on the cylinder = change in kinetic energy of the cylinder

 

[Moetasim]

Hi Peter,

I'm glad to see you're reaching out for help with this problem. Let's break it down step by step.

First, let's define some variables:
F = force applied to the end of the rope (P in the problem)
m = mass of the cylinder (we're given weight, but we need mass for our equations)
r = radius of the cylinder
v = final velocity of the end of the rope
d = distance the end of the rope is pulled

Next, let's list out the given information:
- The cylinder has a weight of 55.0 N (which we'll use to find mass)
- The cylinder has a radius of 0.25 m
- The end of the rope is pulled with a constant force P
- The end of the rope is pulled a distance of 3.00 m
- The end of the rope is moving at a final velocity of 4.00 m/s
- The cylinder rotates without friction (meaning no external torque is applied)

Now, let's think about what equations we can use to solve for P. We know that the final velocity of the end of the rope is related to the rotational speed of the cylinder (since the rope is wrapped around it). We can use the equation v = ωr, where ω is the angular velocity of the cylinder. We can also use the equation α = τ/I, where α is the angular acceleration, τ is the torque applied to the cylinder, and I is the moment of inertia. Finally, we can use the equation I = MR^2 to find the moment of inertia of the cylinder.

Putting all of this together, we can set up the following equations:
v = ωr
α = τ/I
I = MR^2

Now, let's solve for some of the unknowns. First, we can use the weight of the cylinder to find its mass:
m = F/g = 55.0 N / 9.8 m/s^2 = 5.61 kg

Next, we can use the equation v = ωr to find ω, since we know v and r:
ω = v/r = 4.00 m/s / 0.25 m = 16.0 rad/s

Now, we can use the equation α = τ/I to find the torque applied to the cylinder. We know that the rope is wrapped around the cylinder, so