A 6.01-kg box is sliding across the horizontal floor of an elevator. The coefficient of kinetic friction between the box and the floor is 0.291. 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 1.20 m/s2, and (c) accelerating downward with an acceleration whose magnitude is 1.20 m/s2.
F(fr) = μ•N.
1. N=mg => F(fr) = μ•mg
2. N=m(g+a) => F(fr) = μ•m• (g+a),
3. N=m(g-a) => F(fr) = μ•m• (g-a),
To determine the kinetic frictional force that acts on the box in each scenario, we need to first calculate the normal force acting on the box. The normal force is the force exerted by a surface to support the weight of an object resting on it.
In scenario (a), when the elevator is stationary, the normal force is equal to the weight of the box, which can be calculated using the formula:
Normal force = mass * acceleration due to gravity
Normal force = 6.01 kg * 9.8 m/s^2
Normal force ≈ 58.83 N
The kinetic frictional force can be calculated using the formula:
Frictional force = coefficient of kinetic friction * normal force
Frictional force = 0.291 * 58.83 N
Frictional force ≈ 17.11 N
Therefore, the kinetic frictional force that acts on the box when the elevator is stationary is approximately 17.11 N.
In scenario (b), when the elevator is accelerating upward with an acceleration of 1.20 m/s^2, the normal force is modified. The net force acting on the box is the difference between the force of gravity (weight) and the force causing the acceleration of the elevator.
Net force = mass * acceleration
Net force = 6.01 kg * 1.2 m/s^2
Net force ≈ 7.21 N
To find the normal force, we subtract the net force from the force of gravity:
Normal force = weight - net force
Normal force = 6.01 kg * 9.8 m/s^2 - 7.21 N
Normal force ≈ 58.83 N - 7.21 N
Normal force ≈ 51.62 N
The kinetic frictional force can be calculated as before:
Frictional force = coefficient of kinetic friction * normal force
Frictional force = 0.291 * 51.62 N
Frictional force ≈ 15.01 N
Therefore, the kinetic frictional force that acts on the box when the elevator is accelerating upward with an acceleration of 1.20 m/s^2 is approximately 15.01 N.
In scenario (c), when the elevator is accelerating downward with an acceleration of 1.20 m/s^2, the normal force is modified again. This time, the net force acting on the box is the sum of the force of gravity and the force causing the downward acceleration of the elevator.
Net force = mass * acceleration
Net force = 6.01 kg * -1.2 m/s^2 (negative since it is downward)
Net force ≈ -7.21 N
To find the normal force, we add the net force to the force of gravity:
Normal force = weight + net force
Normal force = 6.01 kg * 9.8 m/s^2 + (-7.21 N)
Normal force ≈ 58.83 N + (-7.21 N)
Normal force ≈ 51.62 N
Again, the kinetic frictional force can be calculated:
Frictional force = coefficient of kinetic friction * normal force
Frictional force = 0.291 * 51.62 N
Frictional force ≈ 15.01 N
Therefore, the kinetic frictional force that acts on the box when the elevator is accelerating downward with an acceleration of 1.20 m/s^2 is approximately 15.01 N.
To determine the kinetic frictional force that acts on the box in different scenarios, we need to use the formula:
Frictional force = coefficient of kinetic friction * normal force
The normal force is the force exerted by a surface that is perpendicular to the surface. In this case, since the box is on a horizontal surface, the normal force is equal to the gravitational force acting on the box, which is given by:
Normal force = mass * acceleration due to gravity
where the acceleration due to gravity, g, is approximately 9.8 m/s^2.
(a) When the elevator is stationary, the box does not experience any acceleration, so the normal force is equal to the weight of the box:
Normal force = mass * acceleration due to gravity = 6.01 kg * 9.8 m/s^2 = 58.798 N
Now we can calculate the frictional force:
Frictional force = coefficient of kinetic friction * normal force = 0.291 * 58.798 N = 17.067 N
Therefore, the kinetic frictional force that acts on the box when the elevator is stationary is 17.067 N.
(b) When the elevator is accelerating upward with an acceleration of 1.20 m/s^2, we need to consider the net force acting on the box. In this case, the net force is the difference between the gravitational force and the force applied by the elevator:
Net force = mass * (acceleration due to gravity + acceleration of the elevator) = 6.01 kg * (9.8 m/s^2 + 1.20 m/s^2) = 69.5302 N
The normal force also changes in this scenario because the box is being pushed upward. The equation for the normal force becomes:
Normal force = mass * (acceleration due to gravity + acceleration of the elevator) = 6.01 kg * (9.8 m/s^2 + 1.20 m/s^2) = 69.5302 N
Now we can calculate the frictional force:
Frictional force = coefficient of kinetic friction * normal force = 0.291 * 69.5302 N = 20.209 N
Therefore, the kinetic frictional force that acts on the box when the elevator is accelerating upward with an acceleration of 1.20 m/s^2 is 20.209 N.
(c) When the elevator is accelerating downward with an acceleration of 1.20 m/s^2, the net force acting on the box is the difference between the gravitational force and the force applied by the elevator:
Net force = mass * (acceleration due to gravity - acceleration of the elevator) = 6.01 kg * (9.8 m/s^2 - 1.20 m/s^2) = 47.858 N
The normal force is also affected in this situation because the box is being pushed downward. The equation for the normal force becomes:
Normal force = mass * (acceleration due to gravity - acceleration of the elevator) = 6.01 kg * (9.8 m/s^2 - 1.20 m/s^2) = 47.858 N
Now we can calculate the frictional force:
Frictional force = coefficient of kinetic friction * normal force = 0.291 * 47.858 N = 13.915 N
Therefore, the kinetic frictional force that acts on the box when the elevator is accelerating downward with an acceleration of 1.20 m/s^2 is 13.915 N.