How Is the Smallest Force to Move a Box Up a Ramp Calculated?

• makerfeldt
In summary, to move the box up an incline of 42.0° to the horizontal, the mover must use a force of 87.68 N.
makerfeldt

Homework Statement

A 19.0 kg box sits on a frictionless ramp with a 14.9° slope. The movers pull up a rope attached to the box to move it up. If the rope makes an incline with the ramp that is 42.0° to the horizontal, what is the smallest force F the mover must use to move the box up the incline?

Newton's 2nd Law

The Attempt at a Solution

I was already given the answer to this as 53.8 N, but I can't get there (though I get very close)

I broke it down into 3 forces; the gravitational force, the normal force, and the force the mover exerts on the box. Getting the value 186.2 N for the value of Fgy, I obtained 47.88 N for Fgx by utilizing Fgy divided by cos∅ and then multiplying that value by sin∅. The F exerted by the mover along the x should equal that value of 47.88. Using that value, 47.88 / cos 56.9° (The angle the rope makes with the rotated axis I used) I get a monstrous value of 87.68 N. Any tips or pointers would be much appreciated.

hi makerfeldt! welcome to pf!
makerfeldt said:
A 19.0 kg box sits on a frictionless ramp with a 14.9° slope. The movers pull up a rope attached to the box to move it up. If the rope makes an incline with the ramp that is 42.0° to the horizontal

Using that value, 47.88 / cos 56.9° (The angle the rope makes with the rotated axis I used)

why did you add (14.9° + 42° = 56.9°) ?

The initial angle given for the rope (42) was the angle it makes with the horizontal, but that is not the horizontal I used. I rotated the axis 14.9 so that I could make the ramp's surface the new horizontal. As a result, I reasoned that the new angle the rope would make with the horizontal would be 56.9. Is there a flaw in that reasoning?

Edit: I have seen the flaw in that reasoning, I should have done the opposite.

Thanks for the welcome by the way, I am sure I will be making many posts in the near future :)

1. How does the angle of the ramp affect the speed of the box?

The steeper the angle of the ramp, the faster the box will slide down due to the force of gravity pulling it down.

2. What role does friction play in the box sliding down a ramp?

Friction between the box and the ramp will slow down the box's speed as it slides down. The rougher the surface of the ramp, the more friction there will be.

3. How does the mass of the box affect its acceleration down the ramp?

The mass of the box will affect its acceleration down the ramp. A heavier box will have more inertia and will require more force to accelerate down the ramp compared to a lighter box.

4. Can the height of the ramp affect the distance the box travels?

Yes, the height of the ramp can affect the distance the box travels. The higher the ramp, the longer the distance the box will travel before coming to a stop.

5. What is the relationship between the length of the ramp and the time it takes for the box to reach the bottom?

The longer the ramp, the longer it will take for the box to reach the bottom due to the increased distance the box must travel. However, the angle of the ramp and the mass of the box will also play a role in the time it takes for the box to reach the bottom.

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