# How much work would it take to push a  6 kg  weight up a  4 m  plane that is at an incline of  pi / 4 ?

Apr 17, 2016

$166.31 N m$

#### Explanation:

It is always good to start a mathematical model with certain assumptions. Here, we are ignoring air resistance, friction, and assuming that gravity is a constant $9.8 \frac{m}{s} ^ 2$ downwards.

The force straight down with gravity is given by

$F = m a$

where $m$ is the mass of $6 k g$ and $a$ is acceleration of gravity, $9.8 \frac{m}{s} ^ 2$, so

${F}_{\text{vertical down}} = 6 k g \cdot 9.8 \frac{m}{s} ^ 2 = 58.8 N$.

But since the plane gets in the way and the mass cannot move straight down, it would have to move down the $\frac{\pi}{4}$ plane. Using trigonometry, this gives a force down the plane of

${F}_{\text{down plane}} = 58.8 N \cdot \cos \left(\frac{\pi}{4}\right)$
$= 58.8 \cos 45 = 41.578 N$

In order to move up the plane, you need to exert a force equal to or greater than the force down the plane, which means

${F}_{\text{up plane}} = 41.578 N$

Now, to find the work you use the equation

$W = F d$,

where $F$ is the force we have already found and $d$ is the distance of $4 m$ up the plane, so

$W = 41.578 N \cdot 4 m = 166.31 N m$