A dockworker loading crates on a ship finds that a 23-kg crate, initially at rest on a horizontal surface, requires a 75-N horizontal force to set it in motion. However, after the crate is in motion, a horizontal force of 50 N is required to keep it moving with a constant speed. Find the coefficients of static and kinetic friction between crate and floor.
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To find the coefficients of static and kinetic friction between the crate and the floor, we first need to understand the forces acting on the crate.
1. Static Friction (Fs)
When the crate is at rest, the static friction is the force that opposes the applied force and prevents the crate from moving. We can say that the force of static friction equals the force applied to set the crate in motion.
Fs = 75 N
2. Kinetic Friction (Fk)
Once the crate is in motion, the kinetic friction is the force that opposes its motion and needs to be overcome to keep it moving at a constant speed. We can say that the force of kinetic friction equals the force required to keep the crate moving.
Fk = 50 N
Now, we can use the equations of motion and the definition of friction to find the coefficients of static and kinetic friction.
The equation governing the static friction is:
Fs = μs * Normal Force
The equation governing the kinetic friction is:
Fk = μk * Normal Force
Where:
μs is the coefficient of static friction
μk is the coefficient of kinetic friction
Normal Force is the perpendicular force exerted by the surface on the crate, which equals the weight of the crate (mg).
Now, we can solve for the coefficients of static and kinetic friction.
1. Coefficient of Static Friction:
From the given information, we know that the static friction is 75 N. Since the normal force equals the weight, we have:
Fs = μs * Normal Force
75 N = μs * (23 kg * 9.8 m/s^2)
75 N = μs * 225.4 N
μs = 75 N / 225.4 N
μs ≈ 0.333
2. Coefficient of Kinetic Friction:
From the given information, we know that the kinetic friction is 50 N. Using the same reasoning as above:
Fk = μk * Normal Force
50 N = μk * (23 kg * 9.8 m/s^2)
50 N = μk * 225.4 N
μk = 50 N / 225.4 N
μk ≈ 0.222
Therefore, the coefficients of static and kinetic friction between the crate and the floor are approximately 0.333 and 0.222, respectively.