A 5.10 kg box is sliding across the horizontal floor of an elevator. The coefficient of kinetic friction between the box and the floor is 0.450. Determine the kinetic frictional force that acts on the box for each of the following cases.
(a) The elevator is stationary.
(b) The elevator is accelerating upward with an acceleration whose magnitude is 3.00 m/s2.
(c) The elevator is accelerating downward with an acceleration whose magnitude is 3.00 m/s2.
To determine the kinetic frictional force that acts on the box in each of the given cases, we'll use the following formula:
Frictional force (f) = coefficient of kinetic friction (μ) × normal force (N)
First, let's calculate the normal force acting on the box.
(a) The elevator is stationary:
In this case, the elevator is not moving, so the box is subject to the force of gravity only. Therefore, the normal force (N) is equal to the weight of the box (mg), where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given:
Mass of the box (m) = 5.10 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Normal force (N) = mg = 5.10 kg × 9.8 m/s^2 = 49.98 N
Now, let's calculate the frictional force (f):
Frictional force (f) = μ × N
Given:
Coefficient of kinetic friction (μ) = 0.450
Normal force (N) = 49.98 N
Frictional force (a) = 0.450 × 49.98 N = 22.491 N
Therefore, the kinetic frictional force that acts on the box when the elevator is stationary is 22.491 N.
(b) The elevator is accelerating upward:
In this case, we need to consider both the force of gravity and the acceleration of the elevator.
The normal force (N) is equal to the sum of the weight of the box (mg) and the force exerted by the upward acceleration of the elevator (ma), where m is the mass of the box and a is the acceleration of the elevator.
Given:
Mass of the box (m) = 5.10 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Acceleration of the elevator (a) = 3.00 m/s^2
Normal force (N) = mg + ma
= 5.10 kg × 9.8 m/s^2 + 5.10 kg × 3.00 m/s^2
= 49.98 N + 15.3 N
= 65.28 N
Now, let's calculate the frictional force (f):
Frictional force (f) = μ × N
Given:
Coefficient of kinetic friction (μ) = 0.450
Normal force (N) = 65.28 N
Frictional force (b) = 0.450 × 65.28 N = 29.376 N
Therefore, the kinetic frictional force that acts on the box when the elevator is accelerating upward with an acceleration whose magnitude is 3.00 m/s^2 is 29.376 N.
(c) The elevator is accelerating downward:
In this case, we consider the force of gravity and the acceleration of the elevator, which is in the opposite direction.
The normal force (N) is equal to the weight of the box (mg) minus the force exerted by the downward acceleration of the elevator (ma), where m is the mass of the box and a is the acceleration of the elevator.
Given:
Mass of the box (m) = 5.10 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Acceleration of the elevator (a) = 3.00 m/s^2
Normal force (N) = mg - ma
= 5.10 kg × 9.8 m/s^2 - 5.10 kg × 3.00 m/s^2
= 49.98 N - 15.3 N
= 34.68 N
Now, let's calculate the frictional force (f):
Frictional force (f) = μ × N
Given:
Coefficient of kinetic friction (μ) = 0.450
Normal force (N) = 34.68 N
Frictional force (c) = 0.450 × 34.68 N = 15.606 N
Therefore, the kinetic frictional force that acts on the box when the elevator is accelerating downward with an acceleration whose magnitude is 3.00 m/s^2 is 15.606 N.
To determine the kinetic frictional force in each case, we need to use the formula:
frictional force = coefficient of kinetic friction * normal force
where the normal force is equal to the weight of the box.
(a) When the elevator is stationary, the net force on the box is zero, so the frictional force is equal to the force needed to keep the box from moving. Therefore, the kinetic frictional force is:
frictional force = coefficient of kinetic friction * normal force
In this case, since the box is not moving, the normal force is equal to the weight of the box:
normal force = mass * acceleration due to gravity = 5.10 kg * 9.8 m/s^2 = 49.98 N
Therefore:
frictional force = 0.450 * 49.98 N = 22.49 N
So, the kinetic frictional force when the elevator is stationary is 22.49 N.
(b) When the elevator is accelerating upward, the net force on the box is equal to the sum of the force due to gravity and the force needed to accelerate the box upward. The frictional force opposes the motion of the box, so it acts in the opposite direction to the net force. The net force is given by:
net force = mass * acceleration = 5.10 kg * 3.00 m/s^2 = 15.30 N
The force due to gravity is:
force due to gravity = mass * acceleration due to gravity = 5.10 kg * 9.8 m/s^2 = 49.98 N
Therefore, the normal force is:
normal force = force due to gravity - net force = 49.98 N - 15.30 N = 34.68 N
The frictional force is:
frictional force = coefficient of kinetic friction * normal force
frictional force = 0.450 * 34.68 N = 15.61 N
So, the kinetic frictional force when the elevator is accelerating upward is 15.61 N.
(c) When the elevator is accelerating downward, the net force on the box is equal to the difference between the force due to gravity and the force needed to accelerate the box downward. The frictional force still opposes the motion of the box, so it acts in the same direction as the net force. The net force is given by:
net force = mass * acceleration = 5.10 kg * (-3.00 m/s^2) = -15.30 N
The force due to gravity is still:
force due to gravity = mass * acceleration due to gravity = 5.10 kg * 9.8 m/s^2 = 49.98 N
Therefore, the normal force is:
normal force = force due to gravity + net force = 49.98 N + (-15.30 N) = 34.68 N
The frictional force is still:
frictional force = coefficient of kinetic friction * normal force
frictional force = 0.450 * 34.68 N = 15.61 N
So, the kinetic frictional force when the elevator is accelerating downward is 15.61 N.