Two particles with masses m1 and m2 are joined by a light

Chapter 41, Problem 40

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Two particles with masses m1 and m2 are joined by a light spring of force constant k. They vibrate along a straight line with their center of mass fixed. (a) Show that the total energy 1 2m1u1 2 1 1 2m2u2 2 1 1 2kx 2 can be written as 1 2mu2 1 1 2kx 2 , where u 5 |u1| 1 |u2| is the relative speed of the particles and m 5 m1m2/(m1 1 m2) is the reduced mass of the system. This result demonstrates that the pair of freely vibrating particles can be precisely modeled as a single particle vibrating on one end of a spring that has its other end fixed. (b) Differentiate the equation 1 2mu2 1 1 2kx 2 5 constant with respect to x. Proceed to show that the system executes simple harmonic motion. (c) Find its frequency.