Two identical crates with 100 kg mass are being pushed up at a constant speed an inclined plane that is at the angle of 20 degrees above the horizontal. One plane is frictionless, while the coefficient of friction between the crate and the other incline is 0.20. What is the absolute difference in work needed to push the crate a distance of 2.0 m along each plane?
A) 670 N
B) 1040 N
C) no difference
D) 370 N
E) 134 N

Question

Quizplus · Accepted Answer

The work done by a force is given as W = Fd cos θ, where F is the force applied, d is the distance moved in the direction of the force, and θ is the angle between the force and the direction of motion.

For the frictionless plane, the force required to push the crate up the incline is equal to its weight component parallel to the incline, which is:

F = mg sin θ = 100 kg × 9.8 m/s² × sin 20° ≈ 338.8 N

The work done against the force is:

W = Fd cos θ = 338.8 N × 2.0 m × cos 20° ≈ 1226.7 J

For the plane with friction, the force required to push the crate up the incline is the sum of its weight component parallel to the incline and the force of friction opposing the motion, which is:

F = mg sin θ + μmg cos θ = 100 kg × 9.8 m/s² × sin 20° + 0.20 × 100 kg × 9.8 m/s² × cos 20° ≈ 406.8 N

The work done against the force is:

W = Fd cos θ = 406.8 N × 2.0 m × cos 20° ≈ 1596.4 J

The absolute difference in work is:

|Wfriction - Wfrictionless| = |1596.4 J - 1226.7 J| ≈ 369.7 J ≈ 370 N

Therefore, the absolute difference in work needed to push the crate a distance of 2.0 m along each plane is 370 N, which is answer choice D).

Two Identical Crates with 100 Kg Mass Are Being Pushed