What force must be applied to push a carton weighing 350 N up a 19° incline, if the coefficient of kinetic friction is 0.45? Assume the force is applied parallel to the incline and the velocity is constant. (Enter the magnitude of the force.)

If the carton is moved at uniform velocity ma=0

F-F(fr) –mgsinα = 0
F= F(fr) +mgsinα =μmgcosα+ mgsinα =
=mg (μ•cosα + sinα)

To find the force that must be applied to push the carton up the incline, we need to consider two opposing forces: the force of gravity pulling the carton down the incline and the force of kinetic friction pushing the carton back.

1. Force of Gravity:
The force of gravity can be calculated using the weight of the carton, which is given as 350 N. Since the incline is at an angle of 19°, the force of gravity acting parallel to the incline can be determined by multiplying the weight by the sine of the angle:
Force of gravity parallel to the incline = 350 N * sin(19°).

2. Force of Kinetic Friction:
The force of kinetic friction opposing the motion can be determined by multiplying the coefficient of kinetic friction by the normal force. The normal force can be calculated by multiplying the weight of the carton by the cosine of the angle:
Normal force = 350 N * cos(19°).
Force of kinetic friction = coefficient of kinetic friction * normal force = 0.45 * (350 N * cos(19°)).

3. Net Force Required:
Since the carton is moving up the incline at a constant velocity, the net force acting on the carton must be zero. Thus, the force applied to move the carton up the incline is equal in magnitude but opposite in direction to the combined force of gravity and kinetic friction:
Force applied = Force of gravity parallel to the incline + Force of kinetic friction.

Calculating these values will give us the answer.

Note: The result of the force applied will be negative, indicating the direction opposite to the motion (up the incline).

Let's calculate:

Force of gravity parallel to the incline = 350 N * sin(19°)
Force of kinetic friction = 0.45 * (350 N * cos(19°))
Force applied = - (Force of gravity parallel to the incline + Force of kinetic friction)

Therefore, the magnitude of the force that must be applied to push the carton up the incline is the absolute value of the calculated force applied.