How far must the mass fall to give a rotational rate of 3 rev/s?

[skysunsand]

Homework Statement

A whee and axle having a total moment of inertia of .0002 kg*m^2 is caused to rotate about a horizontal axis by means of an 800g mass attached to a cord wrapped around the axle. The radius of the axle is 2 cm. Starting from rest, how far must the mass fall to give the wheel a rotational rate of 3.0 rev/s?

I changed all the values here to 800g -> 8 kg
2 cm -> .02 m
3 rev/s -> 6pi rad/s

Homework Equations

ωi = 0, ωf= 6pi rad/s

Downward force on the object would be mgh- Ft, but the problem doesn't mention tension at all so I don't know about using that equation.

I can figure out v, if I use v=rw, but I don't know if that's even useful to this problem.

KE total = 1/2 mv^2 + 1/2Iw^2 is likely going to be important, but I have no clue how it would relate to figuring out the height this thing has to fall.

The Attempt at a Solution

^ Above is the thought process of the equations. Otherwise, I haven't got a clue.

 

[anathema]

Hello there,

First of all, great job identifying the equations that could potentially be used to solve this problem. Let's break it down step by step:

1. We know that the rotational rate (ω) is equal to 6π rad/s. We also know that ω = v/r, where v is the linear velocity and r is the radius of the axle. This means that we can find the linear velocity of the mass attached to the cord.

2. Once we have the linear velocity, we can use the equation for kinetic energy (KE = 1/2*m*v^2) to find the initial kinetic energy of the system.

3. We also know that the initial kinetic energy is equal to the final kinetic energy (since the system starts from rest). The final kinetic energy is equal to the rotational kinetic energy (KE = 1/2*I*ω^2), where I is the moment of inertia and ω is the final rotational rate.

4. Now, we can set these two equations equal to each other and solve for the initial linear velocity.

5. Once we have the initial linear velocity, we can use the equation for linear motion (v^2 = u^2 + 2*a*s) to find the distance (s) that the mass must fall.

6. Don't forget to convert the mass from 800g to 8kg and the radius from 2cm to 0.02m.

I hope this helps! Let me know if you need further clarification or assistance. Good luck!