A 67.0 kg crate rests on a level floor at a shipping dock. The coefficients of static and kinetic friction are .800 and .460 respectively.

A) what horizontal pushing force is required to just start the crate moving?

B) What horizontal pushing force is required to slide the crate across the dock at a constant speed.

If anyone could please help, I would greatly appreciate it. THANK YOU!

To solve this problem, we need to consider the forces acting on the crate. There are two types of friction at play: static friction and kinetic friction.

A) To determine the horizontal pushing force required to just start the crate moving, we need to calculate the maximum static friction. The maximum static friction can be found using the formula:

F_static_max = μ_static * N

Where:
F_static_max is the maximum static friction force,
μ_static is the coefficient of static friction, and
N is the normal force acting on the crate.

The normal force N is equal to the weight of the crate, which can be calculated by multiplying the mass of the crate (m) by the acceleration due to gravity (g):

N = m * g

Substituting this into the formula for maximum static friction, we have:

F_static_max = μ_static * (m * g)

Next, we need to determine if the applied force is greater than the maximum static friction force (F_static_max). If it is, then the crate will start moving. So the horizontal pushing force required to just start the crate moving is equal to the maximum static friction force:

F_push = F_static_max

B) To determine the horizontal pushing force required to slide the crate across the dock at a constant speed, we need to calculate the kinetic friction force. The kinetic friction force can be found using the formula:

F_kinetic = μ_kinetic * N

Where:
F_kinetic is the kinetic friction force,
μ_kinetic is the coefficient of kinetic friction, and
N is the normal force acting on the crate (same as before, calculated as N = m * g).

The kinetic friction force opposes the applied force, so the horizontal pushing force required to slide the crate at a constant speed is equal to the kinetic friction force:

F_push = F_kinetic

Now, we have all the necessary information to solve the problem:

Given:
Mass of the crate, m = 67.0 kg
Coefficient of static friction, μ_static = 0.800
Coefficient of kinetic friction, μ_kinetic = 0.460
Acceleration due to gravity, g = 9.8 m/s^2

Step 1: Calculate the normal force acting on the crate:
N = m * g
N = 67.0 kg * 9.8 m/s^2
N ≈ 656.6 N

Step 2: Calculate the maximum static friction force:
F_static_max = μ_static * N
F_static_max = 0.800 * 656.6 N
F_static_max ≈ 525.3 N

A) The horizontal pushing force required to just start the crate moving is approximately 525.3 N.

Step 3: Calculate the kinetic friction force:
F_kinetic = μ_kinetic * N
F_kinetic = 0.460 * 656.6 N
F_kinetic ≈ 301.6 N

B) The horizontal pushing force required to slide the crate across the dock at a constant speed is approximately 301.6 N.