1. To find the coefficient of static friction, we need to analyze the forces acting on the block.
Given:
Mass of the block, m = 57.1 kg
Angle of elevation, θ = 28.3°
Force applied parallel to the slope to start the motion, F = 162 N
First, let's draw the free-body diagram of the block on the slope:
/|
/ |
/ |
/ |
/ |
/ |
m / |
/ θ |
/______|
The component of the force acting parallel to the slope is F_parallel = F * cos(θ):
F_parallel = 162 N * cos(28.3°)
Now, let's analyze the forces acting on the block in the horizontal direction:
ΣFx = F_parallel - fs,max = m * a
where fs,max is the maximum static frictional force and a is the acceleration of the block. At the point when the block just starts moving, a = 0.
Therefore, we have:
F_parallel - fs,max = 0
Rearranging the equation to solve for fs,max:
fs,max = F_parallel
Substituting the value of F_parallel, we get:
fs,max = 162 N * cos(28.3°)
Now, we can relate the maximum static frictional force to the coefficient of static friction using the equation:
fs,max = μs * N
where N is the normal force acting on the block, given by:
N = m * g
Using the value of g ≈ 9.8 m/s², we can calculate N:
N = 57.1 kg * 9.8 m/s²
Substituting the values of fs,max and N into the equation, we get:
162 N * cos(28.3°) = μs * (57.1 kg * 9.8 m/s²)
Now we can solve for the coefficient of static friction, μs.
2. To find the coefficients of static and kinetic friction, we need to use the forces required to set the crate in motion and keep it moving.
Given:
Mass of the crate, m = 25 kg
Force required to set the crate in motion, F_set = 70 N
Force required to keep the crate moving at a constant speed, F_keep = 54 N
The force required to set the crate in motion is equal to the maximum static frictional force (fs,max). Therefore:
fs,max = F_set
Now, let's relate the maximum static frictional force to the coefficient of static friction:
fs,max = μs * N
The normal force, N, acting on the crate is given by:
N = m * g
Using the value of g ≈ 9.8 m/s², we can calculate N:
N = 25 kg * 9.8 m/s²
Substituting the values of fs,max and N into the equation, we get:
F_set = μs * (25 kg * 9.8 m/s²)
Now we can solve for the coefficient of static friction, μs.
To find the coefficient of kinetic friction, we use the force required to keep the crate moving.
The force of kinetic friction, fk, is given by:
fk = μk * N
Using the formula fk = F_keep and the value of N calculated earlier, we can substitute the values into the equation:
F_keep = μk * (25 kg * 9.8 m/s²)
Now we can solve for the coefficient of kinetic friction, μk.