A loaded penguin sled weighing 66 N rests on a plane inclined at angle θ = 24° to the horizontal (Fig. 6-23). Between the sled and the plane, the coefficient of static friction is 0.22, and the coefficient of kinetic friction is 0.13. (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?
To solve this problem, we can break it down into different parts.
(a) To find the minimum magnitude of the force parallel to the plane that will prevent the sled from slipping down the plane, we need to consider the forces acting on the sled. These forces include the gravitational force (weight), the normal force, and the frictional force.
The gravitational force can be calculated using the formula Fg = mg, where m is the mass of the sled and g is the acceleration due to gravity (approximated as 9.8 m/s^2).
The normal force is the perpendicular force exerted by the plane on the sled. In this case, it is equal to the component of the gravitational force that is perpendicular to the plane. So, Fn = Fg * cos(θ).
The frictional force can be calculated using the formula Ff = μs * Fn, where μs is the coefficient of static friction.
To prevent the sled from slipping, the force parallel to the plane (Fp) should be equal to the maximum possible frictional force. The maximum possible frictional force is given by Fmax = μs * Fn.
Substituting the values, we have:
Fmax = μs * Fn
= μs * Fg * cos(θ)
Now, we can substitute the given values:
μs = 0.22
Fg = 66 N
θ = 24°
Fmax = 0.22 * 66 N * cos(24°)
= 0.22 * 66 N * 0.9135 (cosine of 24°)
Calculating this, we get:
Fmax ≈ 13.42 N
Therefore, the minimum magnitude of the force parallel to the plane that will prevent the sled from slipping down the plane is approximately 13.42 N.
(b) To find the minimum magnitude F that will start the sled moving up the plane, we need to consider the coefficient of kinetic friction (μk) instead of the coefficient of static friction.
The formula to calculate the kinetic frictional force is Fk = μk * Fn.
The sled will start moving up the plane when the force Fp is equal to the kinetic frictional force (Fk).
Substituting the given values:
μk = 0.13
Fn = Fg * cos(θ)
Fk = μk * Fg * cos(θ)
= 0.13 * 66 N * 0.9135 (cosine of 24°)
Calculating this, we get:
Fk ≈ 5.68 N
Therefore, the minimum magnitude F that will start the sled moving up the plane is approximately 5.68 N.
(c) If the sled is required to move up the plane at constant velocity, it means that the frictional force is equal to the force Fp pushing the sled up the plane.
Using the same formula as in part (b), Fk = μk * Fp, we can solve for Fp.
Substituting the given values:
μk = 0.13
Fk = 5.68 N
5.68 N = 0.13 * Fp
Dividing both sides by 0.13, we get:
Fp ≈ 43.69 N
Therefore, the value of F required to move the sled up the plane at constant velocity is approximately 43.69 N.