- #1

decentfellow

- 130

- 1

## Homework Statement

A man with mass ##M## has its string attached to one end of the spring which can move without friction along a horizontal overhead fixed rod. The other end of the spring is fixed to a wall. The spring constant is ##k##. The string is massless and inextensible and mantains a constant angle ##\theta## with the overhead rod, even when the man moves. There is friction with coefficient ##\mu## between the man and the ground. What is the maximum distance (in ##\text{m}##) that the man moving slowly can stretch the spring beyond its natural length.

## Homework Equations

I can't seem to figure out what to write in this so as usual I will be writing this ##\vec{F}=m\vec{a}##

## The Attempt at a Solution

Now as the string is constrained to move while making an angle of ##\theta## with the overhead rod so we get the relation

$$T\cos\theta=kx$$

As the man moves slowly it means that he is always in equilibrium. Therefore the man moves till the normal reaction vanishes, i.e. till ##N=0##, because if he moves any further then he would have exert a force on the string this will not let the rope maintain its angle ##\theta## with the overhead rod.

$$T\sin\theta=Mg\implies x=\dfrac{Mg}{k\tan\theta}$$

But on thinking on the answer provided by the book, which is ##x=\dfrac{\mu Mg}{k(1+\mu\tan\theta)}##, I came to conclude that what the book assumes is that that the friction is acting in the opposite direction then that what I had assumed which can be seen in the last figure above. My reasoning, for that direction was that as the man moves rightward there is relative motion with the ground, so the friction resists this(I kinda assumed his slowly walking to be sliding, or how else would he remain in equilibrium).

Also, if the man does move further rightwards from the book's critical point of equilibrium, as the man is always to remain in equilibrium, so he would just generate so much force to be in equilibrium and not let the spring force pull him backwards as he moves rightwards.

So, according to me the man can move only till the normal reaction force vanishes.

Please correct me at the places where my assumptions are wrong.