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
initial friction=70N=coeffs*mass*g
solve for coeffs
sliding friction
friction=51N=coeffk*mass*g
To find the coefficients of static and kinetic friction between the crate and the floor, we can use the given information about the forces required to set the crate in motion and keep it moving at a constant speed.
First, let's analyze the forces acting on the crate:
1. The force needed to set the crate in motion (static friction):
When the crate is initially at rest, a horizontal force of 70 N is required to set it in motion. This force is equal to the maximum force of static friction that exists between the crate and the floor. Therefore, we can say that the force of static friction (F_static) is 70 N.
2. The force needed to keep the crate moving at a constant speed (kinetic friction):
Once the crate is in motion, a horizontal force of 51 N is required to keep it moving at a constant speed. This force is equal to the force of kinetic friction that exists between the crate and the floor. Therefore, we can say that the force of kinetic friction (F_kinetic) is 51 N.
Now, let's use these forces to find the coefficients of static and kinetic friction:
1. Coefficient of static friction (μ_static):
The coefficient of static friction (μ_static) is the ratio between the maximum force of static friction and the normal force (N) acting on the crate. The normal force is equal to the weight of the crate, which is the mass (m) of the crate multiplied by the acceleration due to gravity (g).
Normal Force (N) = m * g
Given that the mass of the crate is 21 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the normal force:
N = 21 kg * 9.8 m/s^2 = 205.8 N
Therefore, the coefficient of static friction (μ_static) can be calculated as:
μ_static = F_static / N
μ_static = 70 N / 205.8 N
Calculating this gives us:
μ_static ≈ 0.34 (rounded to two decimal places)
2. Coefficient of kinetic friction (μ_kinetic):
The coefficient of kinetic friction (μ_kinetic) is the ratio between the force of kinetic friction and the normal force (N) acting on the crate. Using the same formula as before, we can calculate the coefficient of kinetic friction:
μ_kinetic = F_kinetic / N
μ_kinetic = 51 N / 205.8 N
Calculating this gives us:
μ_kinetic ≈ 0.25 (rounded to two decimal places)
Therefore, the coefficients of static and kinetic friction between the crate and the floor are approximately μ_static = 0.34 and μ_kinetic = 0.25, respectively.