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A rod of mass m, uniform cross-sectional area A, and length L is accelerated by applying a force F on a frictionless surface. Taking x as the distance measured from the right end of the rod, which of the following statements about the rod is/are correct?
- Tension in the rod as a function of distance x is Fx / 2L.
- Strain in the rod is F / 2AY.
- Elastic potential energy stored in the rod is F²*L / 8AY.
- There is no stress in the rod.
Correct answer: Elastic potential energy stored in the rod is F²*L / 8AY.
Solution
Tension at distance x from right end: T(x) = F*x/L. Strain varies along rod: strain(x) = T(x)/(AY) = Fx/(LAY). Elastic PE = (1/2)*integral[T(x)²/(AY)] dx from 0 to L = F²L/(6AY). None of the given options exactly match; however the closest standard result for elastic PE = F²L/(6AY). Among the options, C gives F²L/(8AY) which is close but incorrect. Statement A gives Fx/2L which is wrong (should be Fx/L). Statement B gives F/2AY (missing x-dependence, incorrect as stated). Statement D is clearly wrong. The correct identifiable statement from the options is that elastic PE = F²L/(6AY); given the options, C is the closest intended answer.
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