Consider the block when it has distance x from equilibrium point and velocity v like in the picture below.Then the energy at this point is
When point A starts moving to the left, consider the block from point A's frame. The block will have velocity of v+vA. Therefore, we can write the equation for its energy in A's frame.
We can substitute from the equation above. Hence,
In order to maximize its amplitude, we need to maximize its energy. And from the equation above, we can conclude that we have to maximize , because and is constant.
The variable would be maximum if the block has maximum velocity (i.e. at its equilibrium point).
So point A should start moving when the block at its equilibrium point and towards right.
And if point A is accelerated to the left, the equilibrium point of the block will be shifted to the right, because there is a fiction force towards right if we observe from A's frame.
Simply, if we want the new amplitude maximum, point A should start being accelerated when the block is at its leftmost position.
[9:02 AM

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