Horizontal Force Required to move a box up a ramp

[Bahrbarian]

Homework Statement

A block with a mass of m is initially moving up the incline and is increasing speed with an acceleration of a. The applied force F is horizontal. The coefficients of friction between between the block and incline are μs=S and μk=K. The angle of the incline is θ.
What is the magnitude of F?

Homework Equations

I know I am using f=ma. I just am confused as to how all the forces are found, and ofr their directions, and how to find the force of friction.

The Attempt at a Solution

I left all of the terms in just variables so I would learn how to solve this, not just the answer. I have been doing this on some online homework, so I can check my answer, and I don't understand what I am doing wrong. Magnitude of the normal force is equal to F+mg, I think, and then I split that into horizontal and vertical components. And the force of friction should be SxN, which would be Sx(F+mg), I think? I have to eventually calculate what F equals, and what the force of friction equals. I also think that the coefficient for static friction is extraneous information, but I am not positive. Any help would super awesome.

 

[Bandersnatch]

F_g=mg is NOT normal to the incline. Neither is F.
The latter, as said in the problem description, is horizontal. The former... well, think about the direction gravity should be pulling the block. If it were normal, it'd mean that we could walk on walls.
Now take these two forces, and find their normal and parallel to the incline components.
The parallel ones will be responsible for the body moving, the normal ones for the magnitude of the force of friction that opposes the movement(the total normal force times the friction coefficient).
If you have problems with splitting the forces into components correctly, remember that you're building a rectangle with the original force as a diagonal.
You can relate the component forces to mg and F via trigonometry.

 

[Bahrbarian]

I know that F and mg are not not normal to the incline. F is horizontal, mg is vertical. But youre saying that the normal force isn't equal to F+mg? If not, itd be like Fxsinθ+mgcosθ, yes? The magnitude of the normal force, perpendicular to incline, youre saying isn't F+mg?

 

[Bandersnatch]

If not, itd be like Fxsinθ+mgcosθ, yes?

You've got it.
Do the same for parallel forces, add friction to the mix, and see if you can find the answer.

 

[Nickie]

I would approach this problem by first drawing a free body diagram of the block on the incline. This will help visualize all the forces acting on the block and their directions.

The forces acting on the block are the applied force F, the weight of the block mg, the normal force N, and the force of friction Ff. The normal force N is perpendicular to the incline and the force of friction Ff is parallel to the incline and opposite to the direction of motion.

Next, I would write down the equations of motion for the block in the x and y directions. In the x direction, the acceleration is due to the applied force F and the force of friction Ff. In the y direction, the acceleration is due to the normal force N and the weight of the block mg. This can be written as:

ΣFx = F - Ff = ma
ΣFy = N - mg = ma

From these equations, you can solve for the normal force N and the force of friction Ff. The coefficient of kinetic friction is used to calculate the force of friction Ff in this case, since the block is already in motion.

Once you have calculated the normal force N and the force of friction Ff, you can use these values to solve for the applied force F. This can be done by setting the equations of motion in the x direction equal to each other and solving for F. This will give you the magnitude of the horizontal force required to move the block up the ramp.

In summary, the key steps to solving this problem would be:
1. Draw a free body diagram to visualize all the forces acting on the block.
2. Write down the equations of motion in the x and y directions.
3. Solve for the normal force N and the force of friction Ff.
4. Use these values to solve for the applied force F.

I hope this helps and good luck with your homework!