# A balanced lever has two weights on it, the first with mass 3 kg  and the second with mass 37 kg. If the first weight is  9 m from the fulcrum, how far is the second weight from the fulcrum?

Jan 7, 2016

$0.73 m$.

I have added additional explanations to try and make it clearer how the final answer was arrived at.

#### Explanation:

All the above answers look correct to me, but contributors have asked for more explanations etc so I hope my contribution will explain it satisfactorily.

There are 2 conditions necessary for static equilibrium to be achieved :

1. Resultant force must be zero, ie $\sum \vec{F} = 0$.
2. Resultant torque must be zero, ie $\sum \vec{\tau} = 0$.

If we consider the first condition, then it implies that the sum of the 2 downward weight forces must be equal to the upward normal reaction force of the fulcrum on the lever.
Hence the normal reaction force at the fulcrum is given by
$N = 3 g + 37 g = 292 N$.

Considering the second condition, first realize that torque (also called moment of the force) is defined to be the algebraic cross product of the position vector from the axis of rotation (fulcrum) to the applied force, and the applied force itself.
$\vec{\tau} = \vec{r} \times \vec{F}$.

Since torque is a vector, we may assign for example a positive torque to clockwise rotation and negative to anti-clockwise (or vice versa).
(Note that the direction of the torque vector may be obtained from the right hand rule as follows : Curl the fingers of your right hand from the position vector $\vec{r}$ to the direction of the applied force $\vec{F}$ and your thumb will point in the direction of the $\vec{\tau}$.).
Doing so and writing an expression for the second condition for static equilibrium yields :

${\tau}_{1} - {\tau}_{2} = 0$

$\therefore {r}_{1} {F}_{1} - {r}_{2} {F}_{2} = 0$

$\therefore \left(9 \times 3 g\right) - \left({r}_{2} \times 37 g\right) = 0$

$\therefore {r}_{2} = \frac{9 \times 3 \times 9.8}{37 \times 9.8} = 0.73 m$.

Which is exactly the same answer as the other contributors arrived at.

I hope my explanation makes it a bit clearer how the mechanics of it all works. If not, please feel free to ask again and point out precisely which step(s) you don't understand and I will explain in more detail with sketches if required. :)