1. A block of mass 57.1 kg rests on a slope having an angle of elevation of 28.3°. If pushing downhill on the block with a force just exceeding 162 N and parallel to the slope is sufficient to cause the block to start moving, find the coefficient of static friction.
HINT - Draw the block's free-body diagram and apply Newton's second law for the tilted x-component, solving for μs after substituting
fs, max = μsn.
2. A dockworker loading crates on a ship finds that a 25-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 54 N is required to keep it moving with a constant speed. Find the coefficients of static and kinetic friction between crate and floor.
static friction - ???
kinetic friction - ???
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.
1. To find the coefficient of static friction, we need to analyze the forces acting on the block and use the information given. Here's how we can approach this problem step by step:
Step 1: Draw the free-body diagram of the block. This helps us visualize the forces acting on the block. On the diagram, we have the weight force (mg) pointing straight downwards, and the normal force (N) perpendicular to the surface of the slope. Since the block is on a slope, we also have a force of friction acting on it (fs).
Step 2: Apply Newton's second law in the x-direction (parallel to the slope). The force causing the block to move downhill is the applied force parallel to the slope (F_applied) minus the force of friction. The equation can be written as: F_applied - fs = max. Note that we consider only the forces acting parallel to the slope because we are interested in the component that is relevant to causing the block to move.
Step 3: Substitute the expressions for the force of friction and maximum static friction. In this case, the maximum static friction force is given by the product of the coefficient of static friction (μs) and the normal force (N), so we can replace fs with μsN in our equation: F_applied - μsN = max.
Step 4: Use trigonometry to find the normal force. The weight force can be divided into its components, with one component perpendicular to the slope (N) and another component parallel to the slope (mg*sinθ), where θ is the angle of elevation. Since the block is at rest, these two components must balance each other out, so N = mg*cosθ.
Step 5: Substitute the expression for the normal force into the equation from step 3: F_applied - μs(mg*cosθ) = max.
Step 6: Solve for the coefficient of static friction (μs): μs = (F_applied - max) / (mg*cosθ).
Using the given values for F_applied, m, and θ, you can plug them into the equation to calculate the coefficient of static friction (μs).
2. To find the coefficients of static and kinetic friction, we can use the information given about the forces required to set the crate in motion and keep it moving at a constant speed. Here's how to approach this problem:
Step 1: Determine the force of static friction (fs) required to set the crate in motion. Since the crate is initially at rest, the applied force (F_applied) required to overcome static friction is equal to fs. Therefore, fs = F_applied.
Step 2: Determine the force of kinetic friction (fk) required to keep the crate moving at a constant speed. In this case, the applied force (F_applied) is equal to fk because it balances out the force of kinetic friction. Therefore, fk = F_applied.
Step 3: Calculate the coefficients of static and kinetic friction using the equations: μs = fs / N and μk = fk / N, where N is the normal force.
However, to calculate the coefficients of friction, we need the normal force (N). In this case, since the crate is on a horizontal surface, the normal force is equal to the weight force (mg).
Using the given values for F_applied, we can now calculate the coefficients of static and kinetic friction by substituting the respective forces into the equations for μs and μk.