A 9.90-kg box is sliding across the horizontal floor of an elevator. The coefficient of kinetic friction between the box and the floor is 0.372. Determine the kinetic frictional force that acts on the box when the elevator is (a) stationary, (b) accelerating upward with an acceleration whose magnitude is 1.65 m/s2, and (c) accelerating downward with an acceleration whose magnitude is 1.65 m/s2.

To determine the kinetic frictional force acting on the box in different scenarios, we can use the equation:

Frictional force = coefficient of friction × normal force

The normal force acting on an object on a horizontal surface is equal to the weight of the object:

Normal force = mass × acceleration due to gravity

In this case, the acceleration due to gravity is approximately 9.8 m/s^2.

(a) When the elevator is stationary, the box experiences no acceleration. Therefore, the normal force is equal to the weight of the box:

Normal force = mass × acceleration due to gravity = 9.90 kg × 9.8 m/s^2 = 97.02 N

Kinetic frictional force = coefficient of friction × normal force = 0.372 × 97.02 N = 36.07 N

(b) When the elevator is accelerating upward with an acceleration of 1.65 m/s^2, the net force acting on the box is the difference between the force of gravity and the force applied by the acceleration:

Net force = mass × acceleration = 9.90 kg × 1.65 m/s^2 = 16.28 N

The normal force will be slightly reduced due to the upward acceleration:

Normal force = mass × (acceleration due to gravity + acceleration) = 9.90 kg × (9.8 m/s^2 + 1.65 m/s^2) = 113.85 N

Kinetic frictional force = coefficient of friction × normal force = 0.372 × 113.85 N = 42.36 N

(c) When the elevator is accelerating downward with an acceleration of 1.65 m/s^2, the net force acting on the box is the sum of the force of gravity and the force applied by the acceleration:

Net force = mass × acceleration = 9.90 kg × (-1.65 m/s^2) = -16.28 N

The normal force will be slightly increased due to the downward acceleration:

Normal force = mass × (acceleration due to gravity - acceleration) = 9.90 kg × (9.8 m/s^2 - 1.65 m/s^2) = 82.53 N

Since the box is now moving downward, we need to consider the direction of the frictional force. The frictional force will always oppose the relative motion between the box and the floor. Therefore, the kinetic frictional force in this case is:

Kinetic frictional force = coefficient of friction × normal force = 0.372 × 82.53 N = 30.72 N

So, the kinetic frictional forces in the three scenarios are:
(a) 36.07 N (opposite direction of motion)
(b) 42.36 N (opposite direction of motion)
(c) 30.72 N (in the direction of motion)