A 4.51-kg box is sliding across the horizontal floor of an elevator. The coefficient of kinetic friction between the box and the floor is 0.382. 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 1.29 m/s2, and (c) accelerating downward with an acceleration whose magnitude is 1.29 m/s2.

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

frictional force (f) = coefficient of friction (μ) * normal force (N)

The normal force is the force exerted by a surface to support the weight of an object resting on it. In all three scenarios, the weight of the box is the same and can be determined using the formula:

weight (W) = mass (m) * gravitational acceleration (g)

Where:
mass (m) = 4.51 kg (given)
gravitational acceleration (g) = 9.8 m/s²

Now let's calculate the kinetic frictional force for each scenario:

(a) When the elevator is stationary:
In this case, the elevator is not accelerating, so the net force acting on the box is zero. Therefore, the kinetic frictional force must equal the applied force (also zero). Hence, the kinetic frictional force is 0 N.

(b) When the elevator is accelerating upward:
In this scenario, we need to consider the additional force due to the elevator's acceleration. The net force on the box is the difference between the applied force (upward force) and the kinetic frictional force. The applied force is equal to the weight of the box plus the force due to acceleration.

Step 1: Calculate the weight of the box:
W = m * g
W = 4.51 kg * 9.8 m/s²
W = 44.198 N

Step 2: Calculate the force due to acceleration:
force due to acceleration = m * a
force due to acceleration = 4.51 kg * 1.29 m/s²
force due to acceleration = 5.8079 N

Step 3: Calculate the normal force:
In an upward-accelerating elevator, the normal force is given by:
normal force (N) = W - force due to acceleration
N = 44.198 N - 5.8079 N
N = 38.3901 N

Step 4: Calculate the kinetic frictional force:
frictional force (f) = μ * N
f = 0.382 * 38.3901 N
f ≈ 14.681 N

Therefore, the kinetic frictional force when the elevator is accelerating upward is approximately 14.681 N.

(c) When the elevator is accelerating downward:
The process is similar to scenario (b), but this time the force due to acceleration is in the opposite direction.

Step 1: Calculate the weight of the box: (same as before)
W = m * g
W = 44.198 N

Step 2: Calculate the force due to acceleration: (opposite direction)
force due to acceleration = m * a
force due to acceleration = 4.51 kg * 1.29 m/s²
force due to acceleration = 5.8079 N

Step 3: Calculate the normal force: (same as before)
normal force (N) = W + force due to acceleration
N = 44.198 N + 5.8079 N
N = 49.0059 N

Step 4: Calculate the kinetic frictional force: (same as before)
frictional force (f) = μ * N
f = 0.382 * 49.0059 N
f ≈ 18.716 N

Therefore, the kinetic frictional force when the elevator is accelerating downward is approximately 18.716 N.