Net force on ramp

Question

I'm having difficulty understanding the free body diagram for the forces acting on a ramp when a block is sliding down it.

Say a block of mass $m$ is accelerating down a ramp with mass M with inclination $\theta$, where the coefficient of kinetic friction is between the block and the ramp is $\mu_k$. Assume the surface the ramp rests on is frictionless.

I can easily determine the free body diagram of the forces acting on the block:

$$F_{gx}=mg\sin(\theta)$$ $$F_{gy}=mg\cos(\theta)$$ $$F_n=F_{gy}=mg\cos(\theta)$$ $$F_k=\mu_kF_n=\mu_kmg\cos(\theta)$$

But I'm having difficulty establishing the free body diagram and the magnitude of the forces acting on the ramp.

How would both FBD's change if the ramp is now being pushed with a constant force $F$ pointed parallel to the flat surface? Any help would be appreciated.

Answer 1

I will let you figure out the actual projections into the coordinate system of the forces shown above. Just a few things to consider

• Newton's third law: if the incline exerts a force $\color{red}{\bf N}$ on the block, then the block exerts a force $\color{red}{-{\bf N}}$ on the incline. Same goes for the friction force $\color{orange}{{\bf F}_f}$

• There's a normal force $\color{brown}{\bf N''}$ which appears as a consequence of the fact that the incline rests on a surface

• There's no friction force between the incline and the surface it rests on, but you can add in the diagram if needed.

• I also included a external force $\color{green}{{\bf F}_{\rm ext}}$

Answer 2

If the block is sliding down the ramp then the horizontal force on the ramp is.......

$F_{rx} = \cos(\theta)(mg\sin(\theta) - u_kmg\cos(\theta))$