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Erich Poppitz

Lecture 8: Supplementary Notes and Exam- ples Below we give a few examples of other oscillators that undergo SHM. Constrained Spring ▯ A constrained particle on a spring: A particle of mass m is con- strained to move along the x axis. It is attached to a light spring whose other end is ▯xed at point A a distance l from the x axis. The tension in the spring when the particle is at x = 0 is T. Find the frequency of motion for small oscillations. ▯ Answer: The tension in the spring increases as it is stretched, but this turns out not to be important for small oscillations. The new tension is T + ▯T. The force in the x direction is ▯(T + ▯T)sin▯ = ▯(T + ▯T)(x=l) = ▯Tx=l + H:O:T:: (Assuming x=l ▯ 1, the change in tension is only found in the H.O.T.). Thus mx = ▯Tx=l which means that the frequency is q ! = T=(lm): 1 Note that for a small spring l ▯ resting length of spring, T ▯ kl and so q ! = k=m. What would happen if the increase in tension were taken into account? Would you expect the frequency to increase or decrease? Floating Objects ▯ The buoyancy force B is the upward pressure force on a submerged object and is always equal in magnitude to the weight of the uid displaced by the submerged object. Its magnitude is then B = ▯ f s; where ▯ fs the density of the uid and V is the volume of that part of the object that is submerged. The buoyancy force B is a property of the uid and is independent of the object it is acting on. ▯ The weight of an object W is of course downward and its magnitude is W = mg; ~ where m is the mass of the object. The weight W is a property of the object and is independent of the uid. ▯ For a oating object in static equilibrium with a volume V s;esubmerged below water level, the net vertical force is B ▯ W = ▯ V g ▯ mg = 0 (at equilibrium.) e f s;e ▯ For a oating object displaced from equilibrium, the net force is B ▯ W = B + (B ▯ B ) ▯ W = ▯ (V ▯ V )g: e e f s s;e 2 ▯ This net force is nonzero and must be balanced by the acceleration of the object’s CM. If we denote by z the position of the CM, then ▯ f  = m g(Vs▯ V )s;e To get an equation of motion we need to write V ▯Vs s;ein terms of the CM’s position z. ▯ Worked example: ▯nd the period of oscillation for a 2 cm long cylin- drical cork bobbing in water about its vertical axis. The speci▯c gravity of cork is about 0:25. ▯ Solution: For a cyli
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