A cart for hauling ore out of a gold mine has a mass of 407 kg, including its lo

Question

A cart for hauling ore out of a gold mine has a mass of 407 kg, including its load. The cart runs along a straight stretch of track that is sloped 4.07° from the horizontal. A donkey, trudging along and to the side of the track, has the unenviable job of pulling the cart up the slope with a 365-N force for a distance of 201 m by means of a rope that is parallel to the ground and makes an angle of 14.9° with the track. The coefficient of friction for the cart's wheels on the track is 0.0173. Use g = 9.81 m/s2.

Find the work that the donkey performs on the cart during this process.

Find the work that the force of gravity performs on the cart during the process.

Calculate the work done on the cart during the process by friction.

Explanation / Answer

The work done by gravity is equal to the change in potential energy:

W done by gravity = -m*g*h

It's negative because gravity pulls in the opposite direction to the cart's displacement.

where h = (201 m) * sin(4.07°)

so:

W done by gravity = -(407 kg)*(9.81 m/s^2)*(201 m) *sin(4.07°)
W done by gravity = - 5.6959 x 10^4 J

Work done by they donkey equals the distance moved multiplied by the component of the force applied in the direction of movement:

W done by donkey = (201 m) * (365 N) *cos(14.9°)
W done by donkey = 7.0898 x 10^4 J



Work done by friction:

First note that the work done "on the cart by friction" is negative. You know this for two reasons: First, friction tends to slow the cart down rather than speed it up, so it's taking energy from the cart (which is what doing negative work on something means). Secondly, you know that the force of friction is in the opposite direction of the cart's movement. S

The force of friction is the coefficient of friction times the normal force between the cart and the track. Since the track is inclined at 4.07°, the normal force is mg*cos(4.07°). So the magnitude of the force of friction is:

F = - 0.0173 * mg*cos(4.07°)

The negative sign is because force is in the opposite direction of displacement.

When the force is constant and the force is parallel to movement, work equals force times displacement.

W done by friction = F * (201 m)
W done by friction = - 0.0173 * mg*cos(4.07°) * (201 m)
W done by friction = - 0.0173 * (407 kg) * (9.81 m/s^2) *cos(4.07°) * (201 m)
W done by friction = - 1.3848 x 10^4 J

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