[Irodov 1.65] Bar on top of a plank on a frictionless surface with a var. force.

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A plank with a bar on it is analyzed under a growing horizontal force on a frictionless surface. The initial acceleration of both the plank and the bar is the same until the force reaches the maximum static friction limit. Beyond this point, the plank accelerates at a constant rate while the bar continues to accelerate due to the applied force. The discussion highlights the significance of the transition point when the applied force equals the maximum static friction, leading to a change in acceleration dynamics. Graphs illustrating these relationships will provide further insights into the behavior of the system.
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A plank of mass m1 with a bar of mass m2 placed on it lies on a smooth horizontal plane. A horizontal force growing with time t as F — at (a is constant) is applied to the bar. Find how the acceler-ations of the plank w1 and of the bar w2 depend on t, if the coefficient of friction between the plank and the bar is equal to k. Draw the approximate plots of these dependences.
 
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I am discussing this question with a friend.

Initial Thoughts:

* The force is applied to the bar, the plank will experience the same force till a certain point.

* This point is actually when F = at becomes equal to the maximum static friction between the two masses.

* So, initially, till the static friction limit is reached, the bar will move along with the plank as a system, and after that, the plank will have a constant acceleration. While the bar will keep on accelerating.

The graphs will reveal something more. I am curious about the jump when F = km2g
 

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