What would a free-body diagram look like for two objects stacked on top of each other at an angle?
The left side of the bottom object rises say 3 meters, and the right is 4 meters. the top object is reversed at 4m and 3m so that the top part of object two would lie parallel with the floor.
What forces would be present? I know of the more simple forces such as gravity and normal force, but the cos and sin forces are confusing me. any help would be appreciated. thanks.
There isn’t anything more complicated than gravity and normal force. The only one left is friction. Between the two blocks, normal force is perpendicular to the slope, friction is parallel. The direction of friction might be up the slope or 180 degrees opposite down the slope, depending on the motions, if any, of the blocks. Between the bottom of the block and the floor there is normal force perpendicular to the floor and friction parallel, again in either direction.
In a basic inclined plane problem, you usually break the gravity force vector down into sine and cosine parts, but it isn’t necessary. You often re-orient the axes parallel and perpendicular to the slope. That’s helpful because more of the forces are in line with those axes than not. However in this problem, it would just confuse you more since the forces between the bottom block and the floor aren’t going to be aligned with those axes. You’ll get less confused letting up and down be the y axis and left and right be the x axis, not trying to orient your axes along the slope. You will have to break up the normal and friction forces between the two blocks into components, but you can leave gravity and the normal and friction forces between the bottom block and the floor as single components.
For example you’ll break the normal force down into Nx and Ny. Let’s say the blocks are 5m wide. The top of the bottom block makes a right triangle 5m wide and 1m high. The components of the normal force will be in the same ratio, but obviously it is more up than sideways, so the y component is larger. So Ny:Nx = 5:1. Algebraicly:
Ny/Nx = +-5
Use +5 if it’s tipping right, -5 if it’s tipping left.
No need to use sine and cosine, since the triangle is described by the length of its sides, not by its angle. You could find out the angle of N, by doing tan^-1(+-5/1). (Note you might have to add 180 degrees or 2pi radians to get the angle in the right quadrant.) You would be doing a lot of unnecessary work on a calculator. However, you might want to do it that way if it’s more familiar to you. Then you would reduce the two independent variables Nx and Ny to one variable N, with the substitutions
Nx = N cos theta
Ny = N sin theta
If you think about it, taking the cosine of an angle you found by an inverse tangent is going to give you something you could have figured out by geometry. cos theta is going to end up being
cos theta = +-1/sqrt(1^2 + 5^2)
Friction would have a similar ratio, but reversed with the magnitude of the x component being larger.
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