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We are talking about SHO in physics and the forumula for period T of a pendulum was introduced - T=2*pi*(L/g)^.5 where L is the length of the pendulum arm/string and g is acceleration due to gravity. the instructor also said this was an approximation that was only accurate for small angles. so i thought what about large angles? - like if it is dropped from a horizontal position.

I did some work and derived my own general formula for any angle. i dont know how correct it is but here is what i came up with:

T = 2*Theta*[ (2*L)/(g*(1-cos(Theta))) ] ^ (1/2)

Theta is the starting angle that it is dropped from - displaced from the vertical - it is zero at the equilibrium position (just hanging straight down), and it is pi/2 or 90 deg if you hold it out horizontally.

so to test it mathematically against the other formula i made an excel table with values of theta from 0 to pi/2 in increments of 0.1 rad . in one column i had the value from my formula and the next column from the books formula. they were different but i noticed some connections. for example when g=9.8 and L=1 and theta = pi/2, my formula came up with 1.42s and the other came up with 2s. however the books formula is only an approximation for small angles so this is expected. yet when theta = 0.1 there is still discrepency - mine=1.28s , book=2s. I expected that as theta got smaller my formula and the books would have closer and closer numbers, but i was wrong.

then i made a column for percent difference between the two. it ranged from 36 (theta=0.1) to 30 percent (theta=pi/2). however i noticed something interesting. no matter what i changed the length L to, the percent difference column remained constant.

any thoughts?

I did some work and derived my own general formula for any angle. i dont know how correct it is but here is what i came up with:

T = 2*Theta*[ (2*L)/(g*(1-cos(Theta))) ] ^ (1/2)

Theta is the starting angle that it is dropped from - displaced from the vertical - it is zero at the equilibrium position (just hanging straight down), and it is pi/2 or 90 deg if you hold it out horizontally.

so to test it mathematically against the other formula i made an excel table with values of theta from 0 to pi/2 in increments of 0.1 rad . in one column i had the value from my formula and the next column from the books formula. they were different but i noticed some connections. for example when g=9.8 and L=1 and theta = pi/2, my formula came up with 1.42s and the other came up with 2s. however the books formula is only an approximation for small angles so this is expected. yet when theta = 0.1 there is still discrepency - mine=1.28s , book=2s. I expected that as theta got smaller my formula and the books would have closer and closer numbers, but i was wrong.

then i made a column for percent difference between the two. it ranged from 36 (theta=0.1) to 30 percent (theta=pi/2). however i noticed something interesting. no matter what i changed the length L to, the percent difference column remained constant.

any thoughts?

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