A 8.06-kg box is sliding across the horizontal floor of an elevator. The coefficient of kinetic friction between the box and the floor is 0.402. Determine the kinetic frictional force that acts on the box when the elevator is (a) accelerating upward with an acceleration whose magnitude is 2.70 m/s2 and (b) accelerating downward with an acceleration whose magnitude is 2.70 m/s2.
(a) w = normal force on floor = (9.81+2.70)8.06
(b) w = (9.81-2.70)8.06
friction force = .402 w in each case
To determine the kinetic frictional force that acts on the box in this scenario, we need to use the equation for kinetic friction:
Frictional Force = coefficient of kinetic friction * Normal Force
where the normal force is the force exerted by the floor on the box, and it is equal to the weight of the box when the elevator is at rest.
The weight of the box can be calculated using the equation:
Weight = mass * gravity
where the mass of the box is 8.06 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
(a) When the elevator is accelerating upward with an acceleration of 2.70 m/s^2:
To find the normal force, we need to consider the forces acting on the box in the vertical direction. Since the elevator is accelerating upward, the net force in the vertical direction should be equal to the weight of the box minus the force exerted by the floor.
Net force = Weight - Normal Force
Since the box is not moving in the vertical direction, the net force in the vertical direction should be zero:
Net force = 0
This can be written as:
Weight - Normal force = 0
Rearranging the equation, we find:
Normal force = Weight
Using the equation for weight, we can calculate the normal force:
Weight = mass * gravity
Weight = 8.06 kg * 9.8 m/s^2
Once we have the normal force, we can find the kinetic frictional force by multiplying it by the coefficient of kinetic friction:
Frictional Force = coefficient of kinetic friction * Normal Force
Frictional Force = 0.402 * Normal Force
(b) When the elevator is accelerating downward with an acceleration of 2.70 m/s^2:
The same process applies here, but the net force equation in the vertical direction now becomes:
Net force = Weight + Normal Force
Since the box is not moving in the vertical direction, the net force in the vertical direction should be zero:
Net force = 0
This can be written as:
Weight + Normal force = 0
Rearranging the equation, we find:
Normal force = -Weight
Using the equation for weight, we can calculate the normal force:
Weight = mass * gravity
Weight = 8.06 kg * 9.8 m/s^2
Once we have the normal force, we can find the kinetic frictional force by multiplying it by the coefficient of kinetic friction:
Frictional Force = coefficient of kinetic friction * Normal Force
Frictional Force = 0.402 * Normal Force
In summary:
(a) The kinetic frictional force when the elevator is accelerating upward is 31.03 N (0.402 * Normal Force).
(b) The kinetic frictional force when the elevator is accelerating downward is 170.24 N (0.402 * Normal Force).