A loaded penguin sled weighing 73 N rests on a plane inclined at angle θ = 21° to the horizontal (Fig. 6-23). Between the sled and the plane, the coefficient of static friction is 0.24, and the coefficient of kinetic friction is 0.19. (a) What is the minimum magnitude of the force , parallel to the plane, that will prevent the sled from slipping down the plane? (b) What is the minimum magnitude F that will start the sled moving up the plane? (c) What value of F is required to move the sled up the plane at constant velocity?

Well if you multiply the static and the kinetic friction and add gravity to it you get the minimum magnitude.

PhD in Physics

To answer these questions, we need to analyze the forces acting on the penguin sled. There are three key forces involved: gravity (downward), the normal force (perpendicular to the plane), and the force parallel to the plane.

Let's start by breaking down the forces. The weight of the sled (73 N) can be split into two components: one parallel to the plane and one perpendicular to the plane.

1. The component of the weight parallel to the plane is given by:
F_parallel = Weight * sin(θ)
= 73 N * sin(21°)
≈ 24.80 N

2. The component of the weight perpendicular to the plane is given by:
F_perpendicular = Weight * cos(θ)
= 73 N * cos(21°)
≈ 67.49 N

Next, we can find the maximum frictional forces acting on the sled, which depend on the coefficients of friction.

3. The maximum static friction force can be calculated as:
F_static_max = μ_static * F_perpendicular
= 0.24 * 67.49 N
≈ 16.20 N

4. The maximum kinetic friction force can be calculated as:
F_kinetic_max = μ_kinetic * F_perpendicular
= 0.19 * 67.49 N
≈ 12.84 N

Now we can answer the specific questions:

(a) To prevent the sled from slipping down the plane, the force parallel to the plane must exceed the static friction force. Thus, the minimum magnitude of the force required is:
F_min = F_static_max
≈ 16.20 N

(b) To start the sled moving up the plane, the force parallel to the plane must overcome the static friction force. So, the minimum magnitude of the force needed is:
F_min = F_static_max
≈ 16.20 N

(c) To move the sled up the plane at a constant velocity, the force parallel to the plane must balance the kinetic friction force. The magnitude of the force required is:
F = F_kinetic_max
≈ 12.84 N

Note that in all cases, the force parallel to the plane is equal to or greater than the minimum required force.