A dockworker loading crates on a ship finds that a 25 kg crate, initially at rest on a horizontal surface, requires a 84 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.

To find the coefficients of static and kinetic friction between the crate and the floor, we can use the given information about the applied force and the weight of the crate.

Let's start by finding the coefficient of static friction (μs). The coefficient of static friction represents the friction force when the crate is at rest and is just about to start moving.

We know that the applied horizontal force (F_applied) required to set the crate in motion is 84 N. Since there is no vertical acceleration, the normal force (N) acting on the crate is equal to its weight (mg), where m is the mass of the crate and g is the acceleration due to gravity (9.8 m/s^2).

Given:
Mass of crate (m) = 25 kg
Applied force to set it in motion (F_applied) = 84 N
Acceleration due to gravity (g) = 9.8 m/s^2

Weight of crate (W) = mass x acceleration due to gravity = m x g = 25 kg x 9.8 m/s^2 = 245 N

Since the crate is at rest, the static friction force (F_friction_static) is equal to the applied force (F_applied) to set it in motion. Therefore, F_friction_static = F_applied = 84 N.

The formula for static friction is: F_friction_static = μs * N

Plugging in the values, we get:
84 N = μs * 245 N

Simplifying the equation, we find:
μs = 84 N / 245 N ≈ 0.343 (rounded to three decimal places)

So, the coefficient of static friction (μs) between the crate and the floor is approximately 0.343.

Now, let's find the coefficient of kinetic friction (μk). The coefficient of kinetic friction represents the friction force when the crate is already in motion.

We know that the applied horizontal force (F_applied) required to keep the crate moving with a constant speed is 50 N.

Since the crate is moving with a constant speed, the friction force (F_friction_kinetic) opposing its motion is equal to the applied force (F_applied) required to keep it moving. Therefore, F_friction_kinetic = F_applied = 50 N.

Using the formula for kinetic friction: F_friction_kinetic = μk * N

Plugging in the values, we get:
50 N = μk * 245 N

Simplifying the equation, we find:
μk = 50 N / 245 N ≈ 0.204 (rounded to three decimal places)

So, the coefficient of kinetic friction (μk) between the crate and the floor is approximately 0.204.