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  1. If a body is viewed from the frame of another body which is itself accelerating, will work be done by the pseudo force acting on the first body in the frame of second body ( provided the first body is having displacement w.r.t second body)?

  2. Also if the work is done by the pseudo force can we conserve mechanical energy also how do I know that the pseudo force is a conservative force in order to conserve mechanical energy?

Have been thinking about these questions but I am not coming to any conclusion.

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Yes so-called pseudo forces do work and if they were to be describable as a conservative force, then yes the corresponding mechanical energy would be conserved. The best example I can find is the gravitational pull we feel at the surface of the Earth. It is in fact the sum of the "true" gravitational force owing to Newton's law of gravitation and the, latitude dependent, centrifugal force we experience because the Earth is spinning: this gives rise to an effective pseudo-gravitational force whose magnitude depends on the latitude which, for most of our practical purposes, is considered conservative.

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In this case you haven't stated on which body the force is exerted. The second body is accelerating, so I think only that has a net force on it. There is no pseudo force acting on the first body. There must be some contact to have a pseudo force. Also no work is done.

Example: if i move a ball of say 2 grams to a displacement of 10 cm and i see from the ball's reference another ball(heavier, say 8 grams). Work done = 10*2 dynes. By your assumption, if i assume pseudo force is acting on the other ball(which is actually not the case), then work done by you becomes 10*8 dynes. As you can see conservation of energy is not being followed in this as work done by the pseudo force should have been equal to the work done by original force.

So in your case pseudo force doesn't do work.

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Pseudoforce are real forces, i.e., they do everything a normal force would do. The problem with conservative nature is that if the observer is moving in a bizzare fashion so that somehow the curl of its acceleration becomes non zero, then the pseudoforce will be non conservative. For observer moving in one direction, force will be conservative and if you include its potential in mechanical energy then yes, mechanical energy will be conserved.

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  1. The pseudo force will do work (if $F_{pseudo}.d$ is non-zero where d is the displacement of the body). You can also use the Work-Energy theorem to think about this - if the kinetic energy of the body changes (the body is accelerating and the speed of the body is changing) this implies there is work being done and if pseudo force is the only force acting, it will do work.

  2. The condition for any force to be conservative is that the curl of the force vanishes, this is equivalent to the work done in a closed path being zero. So depending upon the problem and not in every case, the pseudo force can be conservative.

In the problem you stated the let's assume that in the lab frame the first body is at rest and the second body is accelerating with the acceleration $a$ along the +x direction (just to simplify the problem, but it could be accelerating in any direction). In the frame of the second body the first body appears to be moving (accelerating with $a$) in the -x direction and so a force (pseudo) must be acting on the first body in the -x direction. This would do work.

Regarding the conservative nature of this force - now let's take $a$ to be $9.8 \ m/s^2$. The second body will observe the first body to accelerate at $a$ along -x direction. Note that the second body will see the entire lab frame (and any object not accelerating with respect to it) accelerate at $a$. If you think about it, this acceleration looks exactly like the acceleration due to a uniform gravitational field. If you imagine moving any body in this situation in a closed path, the work done will be zero (just like in a uniform gravitational field) - hence the pseudo force here is conservative.

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