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

Jan 14, 2016

The explanation below includes two possible approaches to this calculation. The answer (in some ways the least interesting bit!) is $138.6 J$.

#### Explanation:

There are at least two different ways to calculate the answer to this problem.

The first uses the formula $W = F d$, that is, work = force * distance

The distance traveled up the slope will be $4 m$. The required force will be the component of the gravitational force on the object in the direction of motion, $F = m g \sin \theta = 5 \cdot 9.8 \sin \left(\frac{\pi}{4}\right) = 34.5 N$

$W = f \cdot d = 34.5 \cdot 4 = 138.6 J$

The second uses the fact that doing work on something changes its energy. In this case, sliding the object up the ramp increases its height, and therefore its gravitational potential energy.

${E}_{p} = m g h$ where $m$ is the mass $\left(k g\right)$, $g$ is the acceleration due to gravity, $9.8 m {s}^{-} 2$ and $h$ is the height $\left(m\right)$.

Calculating the height uses trigonometry and the definition of sine:

$h = 4 \cdot \sin \left(\frac{\pi}{4}\right) = 2.83 m$

${E}_{p} = m g h = 5 \cdot 9.8 \cdot 2.83 = 138.6 J$ (actually 138.7 on these numbers due to rounding issues)