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 that acts on the box in each scenario, we can use the formula:
Frictional force = coefficient of kinetic friction * normal force
The normal force is equal to the weight of the box, which can be calculated using the formula:
Weight = mass * gravitational acceleration
In this case, the mass of the box is given as 4.67 kg, and the gravitational acceleration is approximately 9.8 m/s^2.
Let's calculate the frictional force in each scenario:
(a) When the elevator is stationary, the box experiences no acceleration. Therefore, the normal force is equal to the weight of the box. Using the formula above, we can calculate the normal force:
Weight = 4.67 kg * 9.8 m/s^2 = 45.716 N (approximately)
Now we can calculate the frictional force:
Frictional force = 0.446 * 45.716 N ≈ 20.392 N
Therefore, when the elevator is stationary, the kinetic frictional force acting on the box is approximately 20.392 N.
(b) When the elevator is accelerating upward with an acceleration of 1.96 m/s^2, the normal force on the box will be greater than its weight. We can calculate the normal force using the formula:
Normal force = mass * (gravitational acceleration + acceleration of the elevator)
Normal force = 4.67 kg * (9.8 m/s^2 + 1.96 m/s^2) ≈ 53.938 N
Now we can calculate the frictional force:
Frictional force = 0.446 * 53.938 N ≈ 24.045 N
Therefore, when the elevator is accelerating upward with an acceleration of 1.96 m/s^2, the kinetic frictional force acting on the box is approximately 24.045 N.
(c) When the elevator is accelerating downward with an acceleration of 1.96 m/s^2, the normal force on the box will be less than its weight. We can calculate the normal force using the same formula as in scenario (b):
Normal force = 4.67 kg * (9.8 m/s^2 - 1.96 m/s^2) ≈ 32.586 N
Now we can calculate the frictional force:
Frictional force = 0.446 * 32.586 N ≈ 14.519 N
Therefore, when the elevator is accelerating downward with an acceleration of 1.96 m/s^2, the kinetic frictional force acting on the box is approximately 14.519 N.