A particle of mass m can slide frictionlessly along a rigid fixed wire, whose shape is shown in the figure below and is given by the equation
y=a+b(x−c)2

Here, a,b and c are positive, real constants.

Raleigh's method

For Raleigh's method, assume that the solution is harmonic, so we assume
x(t)=c+A⋅sin(ωt)

and then substitute into the kinetic and potential energies to find maximum and minimum values. The reason for this asssumption is that x will oscillate around its equilibrium value, which is at the 'bottom' of the parabola in the figure, and the bottom of the parabola happens at x=c. Since we know this, we bake it into our assmption of the solution.

If you assume x(t)=c+A⋅sin(ωt) then you are assuming that x oscillates between c−A and c+A.

Maximum potential energy

Based on your assumption for x(t), find the maximum value of the potential energy of the system. Hint: the sine and cosine functions alone always oscillate between -1 and +1. the square of the sine and cosine functions always oscillate between 0 and 1. If you don't believe this statement, try plotting sin(ωt) or cos2(ωt)for any value of ω.

Vmax=__________

Minimum potential energy

The minimum value of potential energy based on the assumption for x(t) above is given by

Vmin=__________

Maximum kinetic energy

Hint: don't forget that cos2(ωt)+sin2(ωt)=1 and that the maximum value of kinetic energy will happen when potential energy is at its minimum.

The maximum value of kinetic energy based on the assumption for x(t) above is given by:

(NB: use omega in the maple code to represent the greek letter ω. You can preview what your answer looks like by pressing on the button with the magnifying glass.)

Tmax=__________

Minimum kinetic energy

The minimum value of kinetic energy based on the assumtion for x(t) above is given by

Tmin=__________

Raleigh's method for natural frequency

Using Raleigh's method for natural frequency, the natural frequency of this system is given by

(NB: sqrt is the maple function for square root).

ωn=__________