A dockworker loading crates on a ship finds that a 17-kg crate, initially at rest on a horizontal surface, requires a 71-N horizontal force to set it in motion. However, after the crate is in motion, a horizontal force of 57 N is required to keep it moving with a constant speed. Find the coefficients of static and kinetic friction between crate and floor.
To find the coefficients of static and kinetic friction between the crate and the floor, we can make use of the given information and apply Newton's laws of motion.
1. Find the coefficient of static friction:
The coefficient of static friction, denoted by μᵢ, represents the resistance to motion between two objects when they are not moving relative to each other.
The maximum force of static friction (Fᵢₛₜₐₜ) can be calculated using:
Fᵢₛₜₐₜ = μᵢ × N
Where N is the normal force applied by the floor on the crate.
Notice that for the crate to be set in motion, the horizontal force required is 71 N, which is equal to the maximum force of static friction. Therefore, we have:
Fᵢₛₜₐₜ = 71 N
The normal force (N) acting on the crate is equal to the weight of the crate, which can be calculated using:
N = m × g
Where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s²).
Given that the mass of the crate is 17 kg, we have:
N = 17 kg × 9.8 m/s²
N = 166.6 N (approximately)
Substituting the values into the equation for maximum static friction, we get:
71 N = μᵢ × 166.6 N
μᵢ = 71 N / 166.6 N
μᵢ ≈ 0.426
Therefore, the coefficient of static friction between the crate and the floor is approximately 0.426.
2. Find the coefficient of kinetic friction:
The coefficient of kinetic friction, denoted by μₖ, represents the resistance to motion between two objects when they are already in motion relative to each other. It is typically lower than the coefficient of static friction.
The force of kinetic friction (Fₖ) can be calculated using:
Fₖ = μₖ × N
In this case, the horizontal force required to keep the crate moving with constant speed is 57 N, which is equal to the force of kinetic friction. Therefore, we have:
Fₖ = 57 N
Substituting the value of N and the force of kinetic friction into the equation, we get:
57 N = μₖ × 166.6 N
μₖ = 57 N / 166.6 N
μₖ ≈ 0.342
Therefore, the coefficient of kinetic friction between the crate and the floor is approximately 0.342.
To summarize:
- The coefficient of static friction (μᵢ) is approximately 0.426.
- The coefficient of kinetic friction (μₖ) is approximately 0.342.