Suppose that a force F=250 N applied at 20° below the horizontal is just enough to start the motion of the 55 kg crate in Sample Problems 3.5 item 3. Find the (a) normal force and (b) coefficient of static friction. Please explain it by step-by-step with solutions. THANK YOUUU!!
To find the normal force and the coefficient of static friction, we can use the given force and the mass of the crate.
Let's break down the problem step by step:
Step 1: Find the vertical and horizontal components of the applied force.
The vertical component: F_vert = F*sin(20°)
The horizontal component: F_horiz = F*cos(20°)
Step 2: Find the weight of the crate, which is equal to its mass times the acceleration due to gravity.
Weight = mass * gravity
Weight = 55 kg * 9.8 m/s²
Step 3: Calculate the normal force, which is equal in magnitude but opposite in direction to the weight of the crate since the crate is not accelerating vertically.
Normal Force = Weight
Step 4: Determine the frictional force. The frictional force opposes the motion and is related to the normal force by the equation:
Frictional Force = coefficient of static friction * Normal Force
Step 5: Write the equation of motion in the horizontal direction (using Newton's second law):
F_horiz - Frictional Force = mass * acceleration
Step 6: Substitute the known values into the equation and solve for the acceleration:
F_horiz - coefficient of static friction * Normal Force = mass * acceleration
Step 7: Since the crate is just on the verge of moving, the acceleration will be zero. Solve for the coefficient of static friction using the applied force and the normal force:
F_horiz - coefficient of static friction * Normal Force = 0
Now, let's find the solutions:
Step 1: Find the vertical and horizontal components of the applied force.
F_vert = 250 N * sin(20°) = 84.81 N
F_horiz = 250 N * cos(20°) = 232.53 N
Step 2: Calculate the weight of the crate:
Weight = 55 kg * 9.8 m/s² = 539 N
Step 3: The normal force is equal to the weight of the crate:
Normal Force = Weight = 539 N
Step 4: Determine the frictional force:
Frictional Force = coefficient of static friction * Normal Force
Step 5: Write the equation of motion in the horizontal direction:
F_horiz - Frictional Force = mass * acceleration
Step 6: Substitute the known values into the equation and solve for the acceleration:
232.53 N - Frictional Force = 55 kg * acceleration
Step 7: Since the crate is just on the verge of moving, the acceleration is zero, so the equation becomes:
232.53 N - coefficient of static friction * 539 N = 0
Now, solving for the coefficient of static friction (μs):
coefficient of static friction * 539 N = 232.53 N
coefficient of static friction = 232.53 N / 539 N
coefficient of static friction ≈ 0.431
Therefore, the (a) normal force is 539 N and (b) the coefficient of static friction is approximately 0.431.