A dockworker loading crates on a ship finds that a 21-kg crate, initially at rest on a horizontal surface, requires a 70-N horizontal force to set it in motion. However, after the crate is in motion, a horizontal force of 51 N is required to keep it moving with a constant speed. Find the coefficients of static and kinetic friction between crate and floor.

μstatic
μkinetic

To find the coefficients of static and kinetic friction between the crate and the floor, we need to use the given information. We know that the force required to set the crate in motion is the maximum force of static friction, and the force required to keep it moving at a constant speed is the force of kinetic friction.

Let's denote the coefficient of static friction as μ_static and the coefficient of kinetic friction as μ_kinetic.

1. The force required to set the crate in motion is the maximum force of static friction. Mathematically, this can be represented as:

Force of static friction = μ_static * Normal force

Since the crate is on a horizontal surface, the normal force is equal to the weight of the crate. The weight can be calculated using the formula:

Weight = mass * gravitational acceleration

Given that the mass of the crate is 21 kg and the gravitational acceleration is approximately 9.8 m/s^2, we can calculate the weight:

Weight = 21 kg * 9.8 m/s^2 = 205.8 N

Now, we can use the force required to set the crate in motion, which is 70 N, to calculate the coefficient of static friction:

70 N = μ_static * 205.8 N

Dividing both sides of the equation by 205.8 N, we get:

μ_static = 70 N / 205.8 N ≈ 0.34

Therefore, the coefficient of static friction between the crate and the floor is approximately 0.34.

2. The force required to keep the crate moving at a constant speed is the force of kinetic friction. Mathematically, this can be represented as:

Force of kinetic friction = μ_kinetic * Normal force

Again, since the crate is on a horizontal surface, the normal force is the weight of the crate, which is 205.8 N.

Using the force of kinetic friction, which is 51 N, we can calculate the coefficient of kinetic friction:

51 N = μ_kinetic * 205.8 N

Dividing both sides of the equation by 205.8 N, we get:

μ_kinetic = 51 N / 205.8 N ≈ 0.25

Therefore, the coefficient of kinetic friction between the crate and the floor is approximately 0.25.

In conclusion, the coefficients of static and kinetic friction between the crate and the floor are approximately 0.34 and 0.25, respectively.