Bead on a Wire and Harmonic Motion

[NotMrX]

A wire could be shaped like a sinusoidal function and then we could say the bead moves harmonically.

The shape of the wire, so that bead occilates around X = 0.

Y = -50*cos(10X)

If we ignore friction and give it a small dispalcement then it is possible to find angular frequency.

However, when I applied my method to a varation of this problem in a textbook I got a wrong answer.

Part I:
Here was my attempt where Z is the angle from the center of the circle that the bottom of the sinudosoidal function fits on:

S = (r)*(Z) = (50)*(Zmax sin wt)

V= 50*Zmax*w*coswt

V(max) = 50*Zmax*w

Part II:
Energy at the lowest place
E = .5 m (Vmax)^2 - 50mg

Energy at the highest place
E = -50mg*cos(Zmax)

Setting the energies equal:
.5 m (Vmax)^2 - 50g =-50mg*cos(Zmax)

Solving for the velocity:

(Vmax)^2 = 100g*[1-cos(Zmax)]

Part III: combing part I & II
(Vmax)^2 = 100g*cos(Zmax)
(50*Zmax*w)^2 = 100g*[1-cos(Zmax)]

library logged me off, i will finish later

 

[NotMrX]

Nevermind I figured it out. I made mistake in my method before.

s = r Z

r = 1/k where k is the curvature.