# How much work would it take to push a  5 kg  weight up a  15 m  plane that is at an incline of  pi / 4 ?

Feb 9, 2016

The work used to push the weight up the incline is converted into gravitational potential energy (P.E.). When the weight is at the top of the incline its P.E. is equal to the amount of work it took to get it up there. Therefore use P.E. = (mass) x (acceleration due to gravity) x (height of weight) = work done.

#### Explanation:

The work used to push the weight up the incline is converted into gravitational potential energy (P.E.). When the weight is at the top of the incline its P.E. is equal to the amount of work it took to get it up there. Therefore use P.E. = (mass) x (acceleration due to gravity) x (height of weight) = work done.

Use either Pythagoras' theorem ${a}^{2} + {b}^{2} = {c}^{2}$ or trigonometry to find the height at the top of the plane. As the angle is pi/4 (=45 degrees) it's easier to use Pythagoras' theorem because the sides opposite and adjacent to the incline angle are the same. This means:
${\left(\textrm{p l a \ne \le n > h}\right)}^{2} = 2 {\left(\textrm{h e i g h t}\right)}^{2}$
so,
$\textrm{h e i g h t} = \frac{\textrm{p l a \ne \le n > h}}{\sqrt{2}}$

Note that it's preferable to put the surd (the sqrt) in the numerator (the top part of the fraction):
$\textrm{h e i g h t} = \frac{\sqrt{2} \left(\textrm{p l a \ne \le n > h}\right)}{2}$

Now taking:
plane length = 15m,
therefore:
$\textrm{h e i g h t} = \frac{\sqrt{2} \left(15 m\right)}{2}$

acceleration due to gravity is approximately $10 m {s}^{-} 2$ (you may wish to use a more accurate value for this)
and
mass = 5kg
Put these values into:
P.E. = (mass) x (acceleration due to gravity) x (height of weight) = work done
and you will find:
work done = 375 $\sqrt{2}$ joules
which is approximately 530 J.