# When are pseudo forces considered to do work?

I was solving a question in which: there is an inclined wedge with a smaller block on the inclined side of it (all surfaces are smooth). The wedge is then given an acceleration by application of an external force on it, such that the block does not move on the wedge. In this case the smaller block will have 3 forces acting on it:

1. Force of gravity
2. Normal force by the wedge
3. Pseudo force because of the accelerated wedge.

If we were to apply the work-energy theorem for the block, it would be as follows: Work done by gravity + work done by normal + work done by pseudo force = Change in K.E

Now work done by gravity will be zero since the block's displacement is only in horizontal direction (because of the acceleration of the wedge). So to find work done by normal, we must subtract the work done by pseudo force from the change in K.E, right? But in the solution, the work done by pseudo force was not taken into account and the work done by normal force was directly equated to the change in K.E.

Please explain why this was done. Here is the diagram for your reference. Pseudo forces are not a physical effect- they appear when you analyze a system in an accelerating reference frame. If you analyze this system in the lab frame, there simply are no pseudo forces. Naturally, non-existent forces do no work.

If you analyze this system in the accelerating frame, then the block does not move at all and the change in kinetic energy is trivially zero. But assuming you are interested in the change in kinetic energy of the block in the lab frame, you will have to change back to that frame to get the solution.

• Oh ok. So in the lab frame, the only force doing work will be the normal force by the wedge, right? Mar 31 at 7:42
• @MuskaanS Yes, that is correct.
– Chris
Mar 31 at 7:46
• Got it, thanks a lot! Mar 31 at 8:04

You are given the block does not slide on the wedge and the wedge is accelerating in the horizontal direction. Although the block does not move relative to the wedge, in the lab frame the wedge moves, so in this frame the block moves and its change in kinetic energy is due to the net force on the block. You can easily evaluate the work done on the system of block plus wedge, and you can evaluate the work done on the block alone once you determine the forces on the block. In the lab frame (an inertial frame) there are no pseudo (fictitious) forces and as such these forces do no work in an inertial frame The only forces are $$\vec F, \vec N, m\vec g, and \vec f$$, where $$\vec N$$ is the normal force and $$\vec f$$ is the force of friction of the wedge on the block, (assuming no friction between the wedge and the surface it slides on).

In a non-inertial (accelerating) frame, the pseudo forces appear. For a frame fixed on the moving block there is a pseudo-force $$-m \vec a$$ where $$\vec a$$ is the acceleration of the block in the lab frame. The work done on the block in this frame is zero since it is not moving in this frame. The work in the non-inertial frame has to consider the work done by the pseudu-forces as well as the work done by the real forces, and in this case the consideration of the work done by the pseudo-force results in total work done by all forces of zero in the accelerating frame.

In general, a number of different pseudo-forces can appear (e.g., centrifugal and Coriolis). The above is a special case where the accelerating frame is not rotating with respect to the inertial frame.

See the answer by @knzhou to Is work-energy theorem valid in non-inertial frames? on this exchange.