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 YOUU!!
To solve this problem, we first need to break down the force and find the components of the force acting on the crate.
Step 1: Decompose the force
The force F can be decomposed into two components: horizontal and vertical components. The horizontal component (F_x) is given by F_x = F * cos(theta), where theta is the angle below the horizontal. Similarly, the vertical component (F_y) is given by F_y = F * sin(theta).
Given:
F = 250 N (applied force)
theta = 20° (angle below the horizontal)
Calculating the components:
F_x = 250 N * cos(20°)
F_y = 250 N * sin(20°)
Step 2: Calculate the normal force
The normal force (N) is the force exerted by a surface to support the weight of an object resting on it. In this case, if the crate is just about to start moving, the normal force will be equal to the weight of the crate.
We can calculate the weight (W) using the formula W = m * g, where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given:
mass (m) = 55 kg
Calculating the weight:
W = m * g
W = 55 kg * 9.8 m/s^2
Step 3: Determine the direction of motion
In this problem, since the crate is just starting to move, the applied force is equal to the maximum static friction force that opposes the motion. Therefore, the direction of the force F is in the opposite direction of the motion, which is along the positive x-axis.
Step 4: Calculate the coefficient of static friction (μs)
The coefficient of static friction (μs) is the ratio of the maximum static friction force (F_s) to the normal force (N). Therefore, we can solve for μs using the equation μs = F_s / N.
Since the force F is just enough to start the crate's motion, it means F_s = F. So, we can use F_x as the maximum static friction force as F_s acts along the x-axis in the direction opposite to F.
Given:
F_s = F_x
Calculating the coefficient of static friction:
μs = F_s / N
μs = F_x / W
Now, let's calculate the values:
F_x = 250 N * cos(20°)
F_y = 250 N * sin(20°)
W = 55 kg * 9.8 m/s^2
Finally, substitute the values into the equation to find μs:
μs = F_x / W
By following these steps and substituting the given values into the respective equations, you can find the normal force (N) and the coefficient of static friction (μs) for the crate.