Question

A bucket that has a mass of 20 kg when filled with sand needs to be lifted to the top of a 15 meter tall building. You have a rope that has a mass of 0.2 kg/m. Once the bucket reaches the top of the building it only has mass 18 kg because there was a hole in the bottom and sand was leaking out at a constant rate while it was being lifted to the top of the building. Find the work done lifting the bucket (sand and rope) to the top of the building.

Answer 1

Answer:

work done lifting the bucket (sand and rope) to the top of the building,

W=67.46 Nm

Explanation:

in this question we have given

mass of bucket=20kg

mass of rope=$.2\frac{kg}{m}$

height of building= 15 meter

We have to find the work done lifting the bucket (sand and rope) to the building =work done in lifting the rope + work done in lifting the sand

work done in lifting the rope is given as,$W_{1}=Force \times displacement$

=$\int\limits^{15}_0 {.2x} \, dx$ ..............(1)

=$.1\times 15^2$

=22.5 Nm

work done in lifting the sand is given as,$W_{2}=Force \times displacement$

$W_{2}=\int\limits^{15}_0 F \, dx$.................(2)

Here,

F=mx+c

here,

c=20-18

c=2

m=$\frac{20-18}{15-0}$

m=.133

Therefore,

$F=.133x+2$

Put value of F in equation 2

$W_{2}=\int\limits^{15}_0 (.133x+2) \, dx$

$W_{2}=.133 \times 112.5+2\times15\\W_{2}=14.96+30\\W_{2}=44.96 Nm$

Therefore,

work done lifting the bucket (sand and rope) to the top of the building,$W=W_{1}+W_{2}$

W=22.5 Nm+44.96 Nm

W=67.46 Nm