A 21.2-kg box rests on a frictionless ramp with a 14.3° slope. The mover pulls on a rope attached to the box to pull it up the incline. If the rope makes an angle of 41.2° with the horizontal, what is the smallest force F the mover will have to exert to move the box up the ramp?
Can someone explain how to calculate ?
To calculate the smallest force F the mover will have to exert to move the box up the ramp, we need to consider the forces acting on the box.
First, we need to resolve the force of gravity into two components: one parallel to the ramp (mg sinθ) and one perpendicular to the ramp (mg cosθ), where m is the mass of the box (21.2 kg) and θ is the angle of the slope (14.3°).
The perpendicular component (mg cosθ) does not affect the motion along the ramp, so we are only concerned with the parallel component (mg sinθ) in this case.
Next, we need to resolve the force of the rope into two components: one parallel to the ramp (F sinθ') and one perpendicular to the ramp (F cosθ'), where F is the force exerted by the mover and θ' is the angle the rope makes with the horizontal (41.2°).
It is important to note that the force of gravity acting down the ramp is counteracted by the force exerted by the mover pulling the box up the ramp.
Using Newton's second law (F = ma), we equate the net force acting parallel to the ramp to the product of the mass and the acceleration:
F sinθ' - mg sinθ = ma
Since we want to calculate the smallest force F required to move the box up the ramp, we want to find the highest possible acceleration. This occurs when the net force is directed up the ramp, so we neglect the negative sign:
F sinθ' = mg sinθ + ma
From this equation, we can solve for the force F:
F = (mg sinθ + ma) / sinθ'
Lastly, we substitute the given values: m = 21.2 kg, g = 9.8 m/s², θ = 14.3°, θ' = 41.2°. We also know that acceleration down the incline (a) is related to the angle of the slope (θ) by a = g sinθ:
F = (21.2 kg * 9.8 m/s² * sin14.3° + 21.2 kg * 9.8 m/s² * sin14.3° * sin41.2°) / sin41.2°
Calculating this expression will give you the answer, which is the smallest force F the mover will have to exert to move the box up the ramp.