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the inclined plane is accelerating towards the right with a ms^{-2}

As shown in the figure, a body of mass(it is not a ball as in the figure) $m$ is placed on an inclined plane, and the plane is accelerating towards the right. It is also given that $a<g/\mu$.

Now the task is to find the angle at which the particle will be at rest w.r.t inclined plane.

The thing confusing me is which direction will be the force $ma$ acting on the body? Some force should be acting on the plane giving it acceleration $a$ then at the same time body will also get the force right?. But another possibility is as plane moves right body gets force of $ma$ towards left, how is that possible?

Another problem is if the body starts to move down the plane friction $f$ should be acting upwards and downwards otherwise. I tried to check the inequality $a<g/\mu$ as given in the question assuming body is moving downward but I couldn't deduce the inequality from it. How should we approach this problem?

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The thing confusing me is which direction will be the force $ma$ acting on the body?

If the body is at rest with respect to the incline, then it must be only moving (accelerating) horizontally. By Newton's second law, this means that the net force acting on the object is horizontal to the right with no vertical component.

Some force should be acting on the plane giving it acceleration a then at the same time body will also get the force right?

No. One force acts on one body. A force does not act on multiple bodies. The force that acts on the incline does not act on the body. However, the fact that the incline is accelerating "into" the body means there is a force acting on the body from the incline.

But another possibility is as plane moves right body gets force of $ma$ towards left, how is that possible?

As mentioned earlier, the body has a net force to the right.

How should we approach this problem?

You can use Newton's second law recognizing that the acceleration and net force are only acting to the right. $$\sum F_x=ma_x=ma$$ $$\sum F_y=ma_y=0$$ These equations should be sufficient.

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  • $\begingroup$ Okay, but I am getting two different answers for $\theta$ if I assume body had a downward acceleration and body had a upward acceleration, In which cases friction will be uphill and downhill respectively $\endgroup$ – fahd Nov 4 at 16:33
  • $\begingroup$ @fahd With static friction you will have a range of acceptable angles, where the ends of the range are determined by the object starting to slide up or down. $\endgroup$ – Aaron Stevens Nov 4 at 16:41
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The thing confusing me is which direction will be the force ma acting on the body?

There are only two forces on the mass. There is gravity, and there is the normal force from the ramp. The sum of those forces will generate a resultant acceleration.

Some force should be acting on the plane giving it acceleration a then at the same time body will also get the force right?

Not the same force. If I put a piece of paper on a car and push the car, then the paper does not feel the same force that is pushing on the car. In this case, the body will not feel the same force that is pushing on the ramp.

But another possibility is as plane moves right body gets force of ma towards left, how is that possible?

Given the forces on the body, none of them can point left. So the body cannot accelerate to the left. But it is possible that it accelerates to the right more slowly than the ramp. If that happens, then the body will move left relative to the ramp (even while it is accelerating to the right relative to the ground).

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  • $\begingroup$ Even If it has a lesser acceleration, to move uphill it should overcome the force of friction, right? $\endgroup$ – fahd Nov 4 at 16:36
  • $\begingroup$ If it has lesser acceleration, that would necessarily mean friction is not holding it in place. This particular problem doesn't mention friction, so that's just an added complication. If friction is ignored, there's only one angle at which the particle will be at rest. If you consider friction, then there will be a range of angles where it will remain at rest with respect to the ramp. $\endgroup$ – BowlOfRed Nov 4 at 16:46
  • $\begingroup$ youtube.com/watch?v=9vvR54VpK30 please see this guy solving this problem, he is taking a force ma as acting on the body towards left and he considers friction to be upward supporting his initial assumption body moves dow, and he gets the right answer as given exam answer key.. $\endgroup$ – fahd Nov 7 at 18:20

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