# An object with a mass of 8 kg is hanging from an axle with a radius of 27 cm. If the wheel attached to the axle has a radius of 18 cm, how much work would it take to turn the wheel a length equal to the circumference of the axle?

Jan 28, 2016

$W = 6.48 g \pi \textcolor{w h i t e}{x} \text{N}$

#### Explanation:

Here's the picture as I see it:

The circumference of the axle is given by:

${c}_{1} = 2 \pi \times 0.27 = 0.54 \pi \text{m}$

The circumference of the wheel is given by:

${c}_{2} = 2 \pi \times 0.18 = 0.36 \pi \text{m}$

If the wheel turns the circumference of the axle then it must turn through a distance of $0.54 \pi \text{m}$.

How many turns $n$ is this?

$n = \frac{0.54 \cancel{\pi}}{0.36 \cancel{\pi}} = \frac{3}{2}$

The no. of turns the axle completes must be the same so the total distance the axle rotates is:

$\frac{3}{2} \times 0.54 \pi \text{m}$

$= 0.81 \pi \text{m}$

The work done in lifting the 8kg mass is given by:

$W = m g h$

$\therefore W = 8 g \times 0.81 \pi$

$W = 6.48 g \pi \textcolor{w h i t e}{x} \text{J}$