SHM r/R Ratio Question: Solving for Simple Harmonic Motion in a Fixed Trough"

[Saitama]

What is h?

ehild

The solution says that h is the height of centre of mass of ring particle system.

 

[ehild]

The position is stable if the CM gets higher when the system is displaced. I do not see at the moment how that relation comes out.

ehild

 

[TSny]

Btw, is there any trick to do this question? This question is from a test paper and the solution says that for stable equilibrium, the condition is
\frac{1}{h}>\frac{1}{r}-\frac{1}{R}
How did they arrived at this relation?

Use the same type of geometry you have already used to show that the change in height of the CM may be expressed as

$\Delta h = (R-r)(1-cos\theta) - (h-r)[1-cos(\frac{R-r}{r}\theta)]$

Use small angle approximation $cos\beta = 1-\beta^2/2$ to simplify the condition $\Delta h >0$ for small displacement of the hoop.

 

[Saitama]

Use the same type of geometry you have already used to show that the change in height of the CM may be expressed as

$\Delta h = (R-r)(1-cos\theta) - (h-r)[1-cos(\frac{R-r}{r}\theta)]$

Use small angle approximation $cos\beta = 1-\beta^2/2$ to simplify the condition $\Delta h >0$ for small displacement of the hoop.

Thanks TSny, I did reach the relation now but how can I find the least ration using the relation?

 

[ehild]

What is h in terms of r?

ehild

 

[Saitama]

What is h in terms of r?

ehild

Oops, totally missed it, I was so much involved in deriving that equation of TSny that I forgot to use the value of h.

Thanks once again both of you.