A block weighing 71.5 N rests on a plane inclined at 24.1° to the horizontal. The coefficient of the static and kinetic frictions are 0.26 and 0.13 respectively. What is the minimum magnitude of the force F, parallel to the plane, that will prevent the block from slipping?

B)What is the minimum magnitude of F that will start the block moving up the plane?
C)What is the magnitude of F is required to move the block up the plane at constant velocity?

A. F = Fp = 71.5*sin24.1 = 29.20 N.

B. Fn = 71.5*cos24.1 = 65.27 N. = Normal
force = Force perpendicular to the plane

Fs = u*Fn = 0.26 * 65.27 = 16.97 N =
Force of static friction.

F-Fp-Fs = m*a = m*0 = 0
F-29.20-16.97 = 0
F = 46.17 N.

C. Fk = u*Fn = 0.13 * 65.27 = 8.49 N. =
Force of kinetic friction.

F-Fp-Fk = m*a = m*0 = 0
F-29.20-8.49 = 0
F = 37.69 N.

To find the minimum magnitude of the force F that will prevent the block from slipping, start by analyzing the forces acting on the block.

1) Draw a free-body diagram of the block on the inclined plane. We have two forces acting vertically: the gravitational force (mg) pointing downwards and the normal force (N) perpendicular to the plane. We also have two forces acting parallel to the plane: the force F, and the force of friction (f).

2) Resolve the gravitational force and normal force into their components. Since the plane is inclined at an angle of 24.1°, the normal force can be split into two components: N₁ perpendicular to the plane and N₂ acting parallel to the plane. N₁ can be calculated using N₁ = N * cos(θ), where θ is the angle of inclination. N₂ can be calculated using N₂ = N * sin(θ). The gravitational force can be split into two components as well: mg₁ perpendicular to the plane and mg₂ parallel to the plane. mg₁ can be calculated using mg₁ = mg * cos(θ), and mg₂ can be calculated using mg₂ = mg * sin(θ).

3) The force of friction (f) can be calculated using the equation f = μs * N₁, where μs is the coefficient of static friction.

4) For the block to remain at rest and not slip, the force F should be equal to or greater than the force of friction (f). Therefore, the minimum magnitude of F that will prevent the block from slipping is F = f.

5) Calculate F by substituting the values you have into the equation F = μs * N₁.

To find the minimum magnitude of F that will start the block moving up the plane, follow these steps:

1) Again, draw a free-body diagram of the block on the inclined plane. The forces acting on the block are the same as before (gravitational force, normal force, force F, and force of friction).

2) Determine the force of friction (f) using the equation f = μk * N₁, where μk is the coefficient of kinetic friction.

3) For the block to start moving up the plane, the force F should be slightly greater than the force of friction (f). Therefore, the minimum magnitude of F required to start the block moving up the plane is F = f + ε, where ε is a small amount representing a slight increase in force.

To find the magnitude of F required to move the block up the plane at constant velocity, follow these steps:

1) Once again, draw a free-body diagram of the block on the inclined plane. The forces acting on the block are the same as before.

2) Now, since the block is moving up the plane at constant velocity, the force of friction (f) is equal to the force F. Therefore, the magnitude of F required to move the block up the plane at constant velocity is F = f.

3) Calculate F by substituting the values you have into the equation F = μk * N₁.

By following these steps and performing the necessary calculations, you can determine the minimum magnitude of F required for each scenario.