A 6.00 Kg box is sliding across the horizontal floor of an elevator. The coefficient of kinetic friction between the box and the floor is 0.360. Determine the kinetic frictional force that acts on the box when the elevator (a) is stationary (b) accelerating upward with an acceleration whose magnitude is 1.20 m/s squared and (c) accelerating downward with an acceleration whose magnitude is 1.20 m/s squared.

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

Frictional force = coefficient of kinetic friction * normal force

The normal force is the force exerted by the floor on the box, which is equal in magnitude and opposite in direction to the force of gravity.

We can calculate the normal force using the equation:

Normal force = mass * acceleration due to gravity

For the given problem, the mass of the box is 6.00 kg, and the coefficient of kinetic friction is 0.360. The acceleration due to gravity is approximately 9.81 m/s².

(a) When the elevator is stationary:
In this case, the box is not moving, so there is no acceleration. Therefore, the frictional force is the static frictional force.
To calculate the static frictional force, use the equation:

Frictional force = coefficient of static friction * normal force

However, to find the static frictional force, we need to compare it to the maximum static frictional force which can be given by:

Maximum static frictional force = coefficient of static friction * normal force

Since the box is not moving, the maximum static frictional force is equal to the force applied on the box, which is equal in magnitude and opposite in direction to the force of gravity:

Maximum static frictional force = mass * acceleration due to gravity

Plugging in the values, we have:

Maximum static frictional force = 6.00 kg * 9.81 m/s² = 58.86 N

Since the box is not moving, the static frictional force is equal to the maximum static frictional force:

Frictional force (a) = maximum static frictional force = 58.86 N

(b) When the elevator is accelerating upward:
In this case, the net force acting on the box is the force of gravity minus the force exerted by the elevator (which is equal to the mass of the box times its acceleration):

Net force = force of gravity - force exerted by elevator

Force of gravity = mass * acceleration due to gravity

Force exerted by elevator = mass * acceleration

Plugging in the values, we have:

Force of gravity = 6.00 kg * 9.81 m/s² = 58.86 N

Force exerted by elevator = 6.00 kg * 1.20 m/s² = 7.2 N

Net force = 58.86 N - 7.2 N = 51.66 N

The frictional force acts in the opposite direction to the acceleration, so the kinetic frictional force will be in the opposite direction to the net force:

Frictional force (b) = -51.66 N

(c) When the elevator is accelerating downward:
In this case, the net force acting on the box is the force of gravity plus the force exerted by the elevator (which is equal to the mass of the box times its acceleration):

Net force = force of gravity + force exerted by elevator

Force of gravity = mass * acceleration due to gravity

Force exerted by elevator = mass * acceleration

Plugging in the values, we have:

Force of gravity = 6.00 kg * 9.81 m/s² = 58.86 N

Force exerted by elevator = 6.00 kg * 1.20 m/s² = 7.2 N

Net force = 58.86 N + 7.2 N = 66.06 N

Again, the frictional force acts in the opposite direction to the acceleration, so the kinetic frictional force will be in the opposite direction to the net force:

Frictional force (c) = -66.06 N