A 4.67-kg box is sliding across the horizontal floor of an elevator. The coefficient of kinetic friction between the box and the floor is 0.446. 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 1.96 m/s2, and (c) accelerating downward with an acceleration whose magnitude is 1.96 m/s2

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

F_friction = μ * N

Where:
F_friction is the frictional force
μ is the coefficient of kinetic friction
N is the normal force

In each scenario, we need to calculate the normal force first using the equation:

N = m * g

Where:
m is the mass of the box
g is the acceleration due to gravity (approximately 9.8 m/s^2)

Let's calculate the kinetic frictional force for each scenario:

(a) When the elevator is stationary, the box experiences only the force due to gravity, so the normal force is equal to the weight of the box.

N = m * g
N = 4.67 kg * 9.8 m/s^2
N ≈ 45.77 N

Now, we can calculate the frictional force:

F_friction = μ * N
F_friction = 0.446 * 45.77 N
F_friction ≈ 20.41 N

Therefore, when the elevator is stationary, the kinetic frictional force acting on the box is approximately 20.41 N.

(b) When the elevator is accelerating upward with an acceleration of 1.96 m/s^2, the normal force will be altered due to the additional upward force on the box.

N = m * (g + a)
N = 4.67 kg * (9.8 m/s^2 + 1.96 m/s^2)
N ≈ 51.32 N

Now, we can calculate the frictional force:

F_friction = μ * N
F_friction = 0.446 * 51.32 N
F_friction ≈ 22.92 N

Therefore, when the elevator is accelerating upward, the kinetic frictional force acting on the box is approximately 22.92 N.

(c) When the elevator is accelerating downward with a magnitude of 1.96 m/s^2, the normal force will also be altered, but this time reduced due to the additional downward force on the box.

N = m * (g - a)
N = 4.67 kg * (9.8 m/s^2 - 1.96 m/s^2)
N ≈ 34.17 N

Now, we can calculate the frictional force:

F_friction = μ * N
F_friction = 0.446 * 34.17 N
F_friction ≈ 15.22 N

Therefore, when the elevator is accelerating downward, the kinetic frictional force acting on the box is approximately 15.22 N.