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Homework Statement
While going over the exercises in my mechanics tutoring sheet I found that one of the problems' stated answer didn't make sense to me (it's apparently from a textbook):
A weight of mass ##m## is fixed to the middle point of a string of length ##l## as shown in the figure [see attachment] and rotates about an axis joining the ends of the string. The system is in contact with its environment at a temperature ##T##. Calculate the tension ##X## acting between the ends of the string in terms of its dependence upon the distance ##x## between the ends.
The Attempt at a Solution
Let ##\theta## be the angle either string makes with ##x##. By symmetry ##\theta## has to be the same for both strings and so too must the tension. We have ##\cos\theta = \frac{x}{l}## and ##\sin\theta = \frac{2 r}{l}## where ##r## is the radius of the circular trajectory. Then ##2X \sin\theta = \frac{mv^2}{r} = \frac{2mv^2}{l \sin\theta}## so ##X = \frac{mv^2}{l (1 - \cos^2\theta)} = \frac{mv^2 l}{l^2 - x^2} = 2K \frac{l}{l^2 - x^2}## where ##K## is the kinetic energy. Using the equipartition theorem we then have that the average tension in either string is ##\langle X \rangle = k_B T \frac{l}{l^2 - x^2}##.
The answer however is apparently supposed to be ##\langle X \rangle = k_B T \frac{x}{l^2 - x^2}##. But how can this be? This would imply that for ##x = 0## there is no tension in either string which is obviously not true. The tension in either string for ##x = 0## should be ##\frac{mv^2}{l}## as this is exactly half of the centripetal force ##\frac{mv^2}{l/2}##.
My other issue is, the problem statement says "calculate the tension ##X## acting between the ends of the string...". How can someone possibly infer from this that the problem wants the thermal average ##\langle X \rangle## for an ensemble of such systems and not just the actual tension ##X## for anyone copy of the system? Is one to infer it from the lack of specification of the angular velocity ##\omega## of the particle together with the specification of the temperature ##T## of the heat bath?