Linearising compound pendulum equation

[seboastien]

Homework Statement

Linearise T=2pi√(K^2 + h^2)/gh K is known constant

This is a compound pendulum equation, I want to plot some kind of formula with variable T against some kind of formula with variable H in order to find g from the gradient.

Homework Equations

The Attempt at a Solution

so I've got T/2pi all squared times g all substituted to x, h subbed to y and k^2 subbed to constant C and I've got the equation y^2 -yx + C=0 and tried to solve for y=x+β

I've tried implicit differentiation and it's gotten me nowhere

 

[tiny-tim]

hi seboastien!

(try using the X² button just above the Reply box )

Linearise T=2pi√(K^2 + h^2)/gh K is known constant

This is a compound pendulum equation, I want to plot some kind of formula with variable T against some kind of formula with variable H in order to find g from the gradient.

if K is a known constant, can't you make one of the axes √(h² + K²) ?

 

[seboastien]

I would have to make the axis √((h^2 + K^2)/gh ) but that is a good point.

However, I would still like to know how I could linearise it further. I know that a taylor approximation is needed but I don't know how to, or what a value to choose

 

[tiny-tim]

√(1 + (h²/K²) = 1 + (h²/K²)/2 + …

 

[seboastien]

??

 

[tiny-tim]

if h/K is small, then √(1 + (h²/K²)) = 1 + (h²/K²)/2 + …

 

[seboastien]

hmmm, my only issue is that its the sqrt of K^2 + h^2 divided by gh

it also turns out that k is the radius of gyration and I have no scales to measure the pendulum's mass. I believe I need a y=mx + c where the y intercept will be determined by k, g by m, x by T and h by y.

is there any way of achieving this?

 

[tiny-tim]

it also turns out that k is the radius of gyration and I have no scales to measure the pendulum's mass. I believe I need a y=mx + c where the y intercept will be determined by k, g by m, x by T and h by y.

i'm confused …

you said that K was known …

Linearise T=2pi√(K^2 + h^2)/gh K is known constant

 

[seboastien]

That's because I thought I was allowed to measure the pendulums mass.

Don't worry I've worked it out...finally, turns out I've been overcomplicating things.

I'll just plot a graph of h^2 against h*T^2 the y intercept will be -k^2 and the gradient will be g/4pi^2.

Thanks anyway.