# Calculating torque when the lever arm has mass

threecorners

## Homework Statement

A board is set on top of a scaffolding, with dimensions as shown in the diagram. The board weighs 15 kg. A man weighing 70 kg stands on the board as shown. How far out can he stand before the board falls?

## Homework Equations

Torque: $\tau = rFsin(\theta)$
Force due to gravity: $F = mg$

## The Attempt at a Solution

I decided to use calculus, and then combine with the torque equation for the torque the man applies. I am assuming the leftmost point where the scaffolding touches the board is the fulcrum, and the rightmost point only serves to keep the board from rotating clockwise.

First, since gravity is our only force and it points downwards, we have $\tau=rmg$.
Now we use calculus:
$d\tau=rg\cdot dm$, and $dm=\frac{15}{5.5}\cdot dr$, so
$d\tau=\frac{15g}{5.5}r\cdot dr$.

Now I assume the first 1.5m of board on each side of the fulcrum cancel out, so the torque on the right of the fulcrum is $\tau=\int_{1.5}^{4} \frac{15g}{5.5}r\cdot dr=...=18.75g$. I leave the g because I am about to cancel it out.

Now for the lefthand side of my equation, i just need the torque from the man, which will be at a distance of $x$ out, which we must solve for. So
$x\cdot 70\cdot g = 18g$, and so $x\cong 0.27m$.

Okay, that's it for my solution. The book says the answer should be $1.2m$. I haven't done physics in a while but I have done a lot of math, which is why my solution is overly mathy. I assume there's an easier thing to do here that doesn't require calculus like I did...

Homework Helper
Gold Member
Your answer is correct, but try to avoid using calculus when algebra will suffice. The 15 kg mass of the board can be represented by a 15g force acting at the center of mass of the board, that is, acting dead center at 1.25 m from the fulcrum. Thus, 70x = 15(1.25), solve x = .27 m.

I am not sure if you realized that tipping occurs when there is no reaction force at the right support as the board starts to lift off from it.

Looks like the book messed up big time on this one, maybe a calculus error

Wecome to PF!

threecorners
Thank you so much, PhanthomJay! Yeah, wrong answers in a book can really make you question yourself... and yes, center of mass was the magic thing I needed, thanks! One question, what do you mean by:

I am not sure if you realized that tipping occurs when there is no reaction force at the right support as the board starts to lift off from it.

...And thanks again!