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Say I have two blocks on top of each other and the bottom one is accelerated (relative to the ground) with a horizontal acceleration $a$. I would like to understand what the maximum acceleration $a_{max}$ can be such that the top body does not move relative to the bottom one.

If I work in the accelerated frame, I can assume the friction force $F_f = \mu mg$ can balance the fictitious translational force (magnitude $ma$) such that block stays still. This means:

$m a_{max} = m g \mu_{max}$ and thus $a_{max} = g \mu_{max}$.
Where $\mu$ is the coefficient of static friction .

How would I do a similar analysis from an inertial reference frame?

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In the ground frame:
The upper block experiences Friction in the direction of motion of the lower block. This is the only force which acts on the upper block.
so $m a_{max} = m g \mu$ where a is the acceleration of the upper block equal to acceleration of lower block.

PS- The forces on the lower block will be the external force that pulls it, and the friction opposite to the external force from the block above.

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  • $\begingroup$ Thank you. This is exactly my point. From the inertial frame, many physical variables are misleading - for example, friction appears to be doing positive work. Also, in the situation where there is a relative acceleration between the two blocks (because a is too high), you can only compute the acceleration of top using an inertial force and trying to balance it with friction. Or am I missing something here? $\endgroup$ Mar 15, 2018 at 23:45
  • $\begingroup$ From the inertial frame, indeed friction does positive work on the upper block. But in the accelerated frame, no work is done by friction as there is no displacement. So ** Work done by a force is frame dependent. ** Going on in the situation where relative motion between blocks is present, the friction is maximum and it is the only force on the top block, so just apply F=ma. For the lower block External force, which pulls it, is opposed by friction which is again maximum. So again apply, F=ma. $\endgroup$ Mar 16, 2018 at 7:38
  • $\begingroup$ Let's take this further and assume the lower block is at an angle, like a wedge. In this case, if the lower block has acceleration a and I don't want the top one to move relative to it, I get very different answers in: - the inertial frame (top block moving horizontally with same acceleration as the wedge) - the accelerated frame (with a horizontal inertial force with Finer = -ma) This is because the expected motion is either situation is different: up and down the wedge in the inert $\endgroup$ Mar 16, 2018 at 10:41
  • $\begingroup$ This is because the expected motion in either situation is in a different direction: - horizontal in the inertial frame (with acceleration a) - up and down the wedge in the accelerated frame (in equilibrium). The inertial frame version, as I formulated it, seems nonsensical to me. In this frame, the body in the wedge is expected to move forward horizontally with a non-zero acceleration whilst maintained contact (namely through the normal force) with the wedge? $\endgroup$ Mar 16, 2018 at 10:48

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