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I require a clarification on some steps in the solution of the following problem.

Problem: The coefficient of static friction between two blocks shown in figure is μ and the table is smooth.What maximum horizontal force can be applied to the block of mass M so that the blocks move together? (M1=4kg, M2=5kg)problem figure

As the blocks should move together, they should both move with same acceleration. And the acceleration should be w.r.t the same reference point(origin). However I found a solution as shown below: solution

Acceleration of M1 is equal to frictional force only if x1 is a coordinate w.r.t the mass M2. x2 is a coordinate w.r.t some arbitrary origin. How can these two accelerations be equated?

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    $\begingroup$ If the bodies move together their relative acceleration with respect to each other must be zero ( condition for no relative slipping ). $\endgroup$ – Aditya Garg May 18 at 3:51
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Here we can consider different position for frame. But it doesn't change the magnitude or direction of the basis vectors. Thus the rate of change of coordinates w.r.t time will be same in any frame. So is the rate of change of velocity. Thus we can equate the acceleration on arbitrary frames as equal for the blocks to move together.

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