A man pushing a crate of mass
m = 92.0 kg
at a speed of
v = 0.860 m/s
encounters a rough horizontal surface of length
ℓ = 0.65 m
as in the figure below. If the coefficient of kinetic friction between the crate and rough surface is 0.353 and he exerts a constant horizontal force of 277 N on the crate.
To find the work done by the man in pushing the crate across the rough surface, we need to calculate the force of friction first.
1. Calculate the force of friction:
The force of friction can be calculated using the formula:
Force of friction (Ff) = coefficient of friction (μ) * normal force (Fn)
The normal force can be calculated by multiplying the mass of the crate (m) by the acceleration due to gravity (g).
normal force (Fn) = m * g
where g is the acceleration due to gravity, approximately 9.8 m/s².
2. Calculate the normal force:
Fn = m * g
Fn = 92.0 kg * 9.8 m/s²
3. Calculate the force of friction:
Ff = μ * Fn
Ff = 0.353 * (92.0 kg * 9.8 m/s²)
4. Calculate the work done by the man:
The work done, W, is given by the formula:
Work (W) = force applied (Fapplied) * displacement (d) * cos(θ)
In this case, the force applied is the constant horizontal force exerted by the man (277 N) and the displacement is the length of the rough surface (0.65 m). The angle θ between the force and displacement is 0 degrees since they are parallel.
W = Fapplied * d * cos(θ)
W = 277 N * 0.65 m * cos(0°)
Finally, the work done by the man in pushing the crate across the rough surface is the positive difference between the work done by the applied force and the work done by the force of friction:
Total work done = W - work done by friction
Total work done = W - Ff * d
Plug in the calculated values to get the final answer.