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Mine Hoist Problem, Hint #3


DrD

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Mechanics Corner

A Journal of Mechanics & Mathematics by DrD, #54C

 

                                                       Mine Hoist Problem, Hint #3

 

Introduction

The original Mine Hoist Problem was posted 23 August, and it is not 10 October, over 6 weeks later. Thus far, I have not received any attempts at a solution. In this third (and probably last) hint, I want to explore some new ideas and also re-work much of the second hint in vector form. It will be useful to have the previous hints and original problem statement in hand as you read this.

 

 

Hint3C.pdf

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If we look at each point along the curve, we could possibly compute the force Mg*sin(theta) - mg where M is the mass of the wheel and m is the mass to be lifted. This integrated over an infinitesimal unit of arc length would give us work done , say W1 . W1 -  Iw^2  /2  (I  is the moment of inertia of the wheel and w is the angular velocity of the wheel) = Mv^2 / 2 (where v is the velocity of center of mass of wheel). The velocity of the wheel will be the velocity of the rope and the velocity with which mass m is hoisted up since they are connected.  Since the wheel does not slip, the angular velocity w and velocity of center of mass v can be linked.  

Possibly, the force (M*g*sin(theta) - mg) can be  connected to the contact point velocity ???

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