A block of mass m lies on the rough horizontal floor of a train accelerating rightward with acceleration a. The block is attached to a horizontal spring of force constant k. A positive extension x makes the spring pull the block rightward. The coefficient of static friction is μ and a > μg. For the block to remain stationary relative to the train, x must satisfy:
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Question

A block of mass m lies on the rough horizontal floor of a train accelerating rightward with acceleration a. The block is attached to a horizontal spring of force constant k. A positive extension x makes the spring pull the block rightward. The coefficient of static friction is μ and a > μg. For the block to remain stationary relative to the train, x must satisfy:

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G1 Physics · Accepted Answer

In the train frame, pseudo force ma acts leftward and the spring force kx acts rightward. Static friction must supply f = ma-kx, with |f|≤μmg. Therefore |ma-kx|≤μmg. This gives ma-μmg ≤ kx ≤ ma+μmg, hence m(a-μg)/k ≤ x ≤ m(a+μg)/k.