A mass m is attached to the end of a light rod (length L), which is attached to

Question

A mass m is attached to the end of a light rod (length L), which is attached to a fixed point A. As shown in the diagram below, the mass is also attached to a spring, of spring constant k and natural length b. The other end of the spring (C) is free to move along a horizontal frictionless track, such that the spring remains vertical at all times. (a) Find the total potential energy of the mass m (with contributions from both graviational potential and elastic potential energy from the spring as a function of the angle that the rod makes with the vertical axis. (b) Find the angles corresponding to any points of equilibrium, and discuss their stability (including any dependence on the parameters m, k, L, b)

Explanation / Answer

a] Let gravitationakl potential energy at A be zero.

total potential energy at a location E = - mg L cos theta + 0.5 k(L cos theta - b)^2

b] For equilibrium, potential energy should be minimum, so dE/d theta = 0

mgL sin theta + 2*0.5 k ( L cos theta -b)*-L sin theta = 0

mg = k (Lcos theta - b]

mg/k = L cos theta - b

L cos theta = b+ mg/k

cos theta = (b+ mg/k)/L

theta = arccos(b/L + mg/kL). answer

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