Find the magnitude of small oscillations

[prabhat rao]

Homework Statement

a rope is tied between 2 walls as shown.a bead of mass 'm' is on the rope as shown. it is constrained to move in the horizontal direction. it is tied to a spring of force constant 'k'- N/m. the spring is initially at its free length 'H'. the bead is displaced by a small displacement 'x' in the horizontal direction. does it execute SHM.If so find the magnitude of small oscillations?
no friction.


the figure is attached!

Homework Equations

T =2 pie/omega

The Attempt at a Solution

Consider the spring to make an angle q with the vertical
The mass in equilibrium in the y direction at all the times
Fsin q = mg
F (h/l)=mg
F = mgl/h
-Fcosq = f_restoring
-Fx/l = f_restoring
-mgl/hl *x =f_restoring
-mgx/h = f_restoring
-mgx/h = ma
ma+mgx.h = 0
a differential equation
omega = sqrt (g/h)
T = 2 pie * sqrt (h/g)
Now the answer is dimensionally correct

method 2

Since the force exerted by the spring is the vectorial sum of the forces along both the directions
F_y/(F_x) = tan q
-F_x= f_restoring = F_y/(tanq)
F_y intially is mg
f_restoring = -mgx/h
so this would be give

T= 2 pie *sqrt (h/g)

An amazing result independent of the spring constant of the force
A spring can only influence the motion along the direction of the spring

Is the solution?? if yes can anybody explain what it means

thank you

 

[variation]

Hello,

I have a result which shows that the motion is NOT a simple harmonic moiton.
Because the net force on the bead is proportional to $$x^3$$.
This is just my opinion.


Regards

 

[prabhat rao]

Can u please give your method

 

[Office_Shredder]

An amazing result independent of the spring constant of the force

This is the first sign something is wrong. Looking at your equations, I can't find anything that actually states the force the spring acts on the object (if it's there, it's certainly missing the spring constant

 

[prabhat rao]

Is the motion simple harmonic??Please help me

 

[Office_Shredder]

I'll give you a hint

F_y intially is mg

Wrong! There's a spring pulling on it too