A loaded penguin sled weighing 81 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 0.13. (a) What is the minimum magnitude of the force (F), 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?

To find the minimum magnitude of the force (F) required in each scenario, we can use the concept of static and kinetic friction.

(a) To prevent the sled from slipping down the plane:
The maximum static friction force can be calculated using the formula: fs ≤ μs * N
where fs is the static friction force, μs is the coefficient of static friction, and N is the normal force acting on the sled.

The normal force can be calculated as N = mg * cos(θ)
where m is the mass of the sled and g is the acceleration due to gravity.

Given that the sled weighs 81 N, we can calculate its mass by dividing the weight by the acceleration due to gravity (m = 81 N / 9.8 m/s^2 ≈ 8.27 kg).

Now:
Normal force, N = 8.27 kg * 9.8 m/s^2 * cos(21°)

The maximum static friction force is:
fs ≤ 0.24 * N

The minimum magnitude of the force (F) required to prevent slipping down the plane is equal to the static friction force:
F = fs

Substituting the values, we can calculate F.

(b) To start the sled moving up the plane:
The force required to start motion is equal to the force of kinetic friction.

The maximum kinetic friction force can be calculated using the formula: fk = μk * N
where fk is the kinetic friction force and μk is the coefficient of kinetic friction.

The minimum magnitude of the force (F) required to start the sled moving up the plane is equal to the kinetic friction force:
F = fk

Substituting the given values, we can calculate F.

(c) To move the sled up the plane at a constant velocity:
If the sled is already moving at a constant velocity, the friction force acting on it is equal to the force applied in the opposite direction.

Therefore, to move the sled up the plane at constant velocity, the force applied (F) must overcome the kinetic friction force:

F > fk

Substituting the given values, we can calculate F.

By performing these calculations, you will find the minimum magnitude of forces required in each scenario.