A dock worker loading crates on a ship finds that a 29 kg crate, initially at rest on a horizontal surface, requires a 90 N horizontal force to set it in motion. However, after the crate is in motion, a horizontal force of 60 N is required to keep it moving with a constant speed. The acceleration of gravity is 9.8 m/s^2

Find the coefficient of static friction be- tween the crate and floor.

See previous post.

To find the coefficient of static friction between the crate and the floor, we need to consider the forces acting on the crate.

When the crate is initially at rest, it requires a 90 N horizontal force to set it in motion. This force is equal to the force of static friction between the crate and the floor. Let's call this force Fs.

Using Newton's second law, we know that force is equal to mass times acceleration: F = ma.

Given that the mass of the crate is 29 kg and the acceleration is unknown, we can substitute these values into the equation: Fs = 29 kg * a.

Now, when the crate is in motion and has a constant speed, a horizontal force of 60 N is required to keep it moving. At this point, the force of static friction is no longer applicable as the crate is no longer stationary. Instead, we need to consider the force of kinetic friction, which is denoted as Fk.

The force of kinetic friction is given by the equation Fk = μk * normal force, where μk is the coefficient of kinetic friction and normal force is the force perpendicular to the surface. In this case, the normal force is equal to the weight of the crate, which is mass * acceleration due to gravity: normal force = 29 kg * 9.8 m/s^2.

Since the force required to keep the crate moving is 60 N and the force of kinetic friction is equal to this force, we have: Fk = 60 N = μk * (29 kg * 9.8 m/s^2).

Now we have two equations: Fs = 29 kg * a and Fk = 60 N = μk * (29 kg * 9.8 m/s^2).

To find the coefficient of static friction (μs), we need to find the maximum value of static friction (Fs) and divide it by the normal force.

Since we have the mass of the crate (29 kg) and the force required to set the crate in motion (Fs = 90 N), we can substitute these values into the equation: Fs = μs * (29 kg * 9.8 m/s^2).

Solving for μs, we have: μs = Fs / (29 kg * 9.8 m/s^2).

Substituting the given values, μs = 90 N / (29 kg * 9.8 m/s^2).

Calculating this expression, we find that the coefficient of static friction (μs) between the crate and the floor is approximately 0.317.