physics homework help

Question

A crate of mass 15kg is pulled by a massless rope up a 36.9 degree ramp. The rope passes over an ideal pulley and is attached to a hanging crate of mass 16.40kg. the crate moves 1.54m, starting from rest. Find the work done by gravity on the sliding crate in Joules.

Answer 1

Let

m = 15 kg be the mass of the crate on the ramp,

M = 16.40 kg be the hanging mass,

α = 36.9° be the angle of the ramp,

d = 1.54 m be the distance the crate on the ramp moves up,

g = 9.81 m/s² be gravitational acceleration,

W (to be computed) be the work of gravity,

ΔPE be the change in potential energy of both masses.

First by definition of potential energy

ΔPE = -W

Second, the potential energy of the system is the sum of the potential energies of the two masses.

The approach to this problem is then to calculate the change in potential energy

for each mass; their sum then leads to the work of gravity.

Let's calculate the change in potential energy of the crate on the ramp.

As the crate moves along the ramp distance d, its change in height is

dsin(α)

With the crate moving up, the change in potential energy is positive, and is

mgdsin(α)

The hanging crane moves down the distance d.

Its change in potential energy is negative, and is

-Mgd

Putting it all together

W = -ΔPE = -(mgdsin(α) - Mgd) = gd(M - msin(α))

Substituting actual numbers

W = 9.81×1.54×(16.40 - 15×sin(36.9°)) = 111.7 J