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 each scenario, we can use the formula:
Frictional force = coefficient of kinetic friction * normal force
The normal force is the force exerted by the floor on the box perpendicular to the surface. In each scenario, the normal force can be calculated using Newton's second law:
Normal force = mass * acceleration
Let's calculate for each scenario:
(a) When the elevator is stationary, the box is not accelerating. Therefore, the normal force is equal to the gravitational force acting on the box:
Normal force = mass * acceleration due to gravity
Normal force = 9.90 kg * 9.8 m/s^2
Normal force = 97.02 N
Now, we can calculate the frictional force:
Frictional force = coefficient of kinetic friction * normal force
Frictional force = 0.372 * 97.02 N
Frictional force = 36.02 N
Therefore, when the elevator is stationary, the kinetic frictional force acting on the box is 36.02 N.
(b) When the elevator is accelerating upward with an acceleration of 1.65 m/s^2, the normal force will be greater than the gravitational force:
Normal force = mass * (acceleration due to gravity + acceleration of the elevator)
Normal force = 9.90 kg * (9.8 m/s^2 + 1.65 m/s^2)
Normal force = 118.26 N
Frictional force = coefficient of kinetic friction * normal force
Frictional force = 0.372 * 118.26 N
Frictional force = 43.99 N
Therefore, when the elevator is accelerating upward, the kinetic frictional force acting on the box is 43.99 N.
(c) When the elevator is accelerating downward with an acceleration of 1.65 m/s^2, the normal force will be less than the gravitational force:
Normal force = mass * (acceleration due to gravity - acceleration of the elevator)
Normal force = 9.90 kg * (9.8 m/s^2 - 1.65 m/s^2)
Normal force = 87.93 N
Frictional force = coefficient of kinetic friction * normal force
Frictional force = 0.372 * 87.93 N
Frictional force = 32.68 N
Therefore, when the elevator is accelerating downward, the kinetic frictional force acting on the box is 32.68 N.
Note: The gravitational force and acceleration due to gravity are considered positive in these calculations, so they are not displayed with a negative sign.