A torsion bar is made from a material of uniform properties· The angular twist Θ of one end of…

A torsion bar is made from a material of uniform properties· The angular twist Θ of one end of the shaft relative to the other is a function of the torque L applied to the bar and the temperature T. Show that, if the angular twist of the shaft is held constant as the shaft is heated, the decrease in torque per unit increase in temperature has a value which is exactly equal to the isothermal variation of the entropy S of the shaft with θ.

he torsional stiffness of the bar is defined as  and is a function of T and θ. Show that, at constant θ, the change

of torsional stiffness with temperature is equal to

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