A mass m is placed on top of a frictionless surface and attached to two slinky s

Question

A mass m is placed on top of a frictionless surface and attached to two slinky springs (information from wikipedia) with different constants k1 and k2 (See figure below) separated by a distance a. Assume that their natural length is negligible and ignore their masses. a) Draw the free body diagram and find the equilibrium position xo (in terms of a). b) Show that when the initial condition is moved away from the equilibrium position , the equation of motion reduces to that of a harmonic oscillator. (Need hint?) c) Find the angular frequency of the oscillations. d) Determine the frequency for the following limiting cases: (i) m right arrow infinite (ii) k1= k2; (iii)k1 right arrow 0.

Explanation / Answer

For a mass undergoing simple harmonic oscillator, the displacement of the mass as a function of time is given as:

x(t) = A*sin(?*t) [to within a phase factor]

where ? = sqrt(k/m), k being the force constant for the restoring force, and m being the mass.

The velocity of oscillating mass is given by the time derivative of the position:

v(t) = A*?*cos(?*t)

The potential energy of the oscillator is given by:

U(t) = (k/2)*(x(t)

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