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Where does the equation for simple pendulum motion come from? \[T=2\pi \sqrt{\frac{ l }{ g }}\] where T is period, l is string length, g is acc. due to gravity
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akash123
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write down the equation of motion for pendulum
akash123
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|dw:1349777134567:dw|
apple_pi
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why is the mg force not straight down?
akash123
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sorry...mg is straight down only
akash123
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does it help u in finding the time period?
apple_pi
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I want to know how the formula is derived...
akash123
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I know
write down the equation of motion along the tangent...
m( a_t) = mg sin(theta)
a_t---tangential acc
akash123
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it's fine?
akash123
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net force along tangent is mg sin (theta)
apple_pi
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yes
akash123
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now v_tangential= r ( angular vel)
akash123
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differentiate it
a_tangential= r(angular acc)
apple_pi
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ok
akash123
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r is the length of pendulum in this case
akash123
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now substitute a_tangential into
m( a_t) = mg sin(theta)
apple_pi
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m(r*angular_acc) = m(sin(theta))
akash123
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m ( l(angular acc))= mg sin(theta)....l=r= length of pendulum
akash123
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now l(angular acc))= g sin(theta)
apple_pi
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where did the l come from?
akash123
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l=r=length of pendulum...pendulum is like bob is moving in a circle pivoted at centre
apple_pi
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oh right... l = r. I forgot about that
akash123
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now acceleration = g/l sin(theta)
apple_pi
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yep
akash123
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equation of SHM is
a= - w^2 x
then time period T= 2 pi/w
apple_pi
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I havent learnt SHM yet.
whats w?
akash123
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w(omega)---angular frequency
acceleration = g/l sin(theta)
for small theta
acceleration = g/l (theta) sin(theta)= theta
akash123
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u know wave...
omega= 2 pi (nu)
apple_pi
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yep
akash123
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so we say SHM is a periodic motion in which acceleration is directed towards a fixed point which is called equilibrium point.
apple_pi
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ok. I know the basics of it
akash123
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ok..
so acceleration = g/l (theta)
akash123
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for small theta
apple_pi
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yep
akash123
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now compare this with
a= - w^2 x
where x is the displacement from fixed point
apple_pi
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so for small angle x = theta
akash123
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so acceleration and displacement are in opposite direction
apple_pi
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yes seems logical
akash123
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here theta is x....
for small angle
sin(theta)= theta
akash123
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so in the case of pendulum
angular displacement and angular acceleration are in opposite direction
akash123
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write six in the form a power series
sin x= x - x^3/3! + x^5/5!-------
akash123
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so for small x
sin x =x
akash123
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neglect higher order terms
apple_pi
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aha
akash123
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so we have
acceleration =- g/l (theta)
apple_pi
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yes
akash123
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so u have w^2= g/l
akash123
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so T= 2 pi/w
apple_pi
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I forgot what w is equal to
akash123
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w= angular frequency= 2 pi( frequency)
apple_pi
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so T = 2pi/2pi?
akash123
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T= (2 pi)/ (2 pi( frequency))
akash123
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:)
akash123
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so T= 1/ (frequency)
apple_pi
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so where does the √ come in?
akash123
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w^2= g/l
so what will be w?
apple_pi
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√(g/l)
akash123
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so T=?
apple_pi
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2π
T=-----
√(g/l)
akash123
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so u got the time period.
apple_pi
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=2π√(l/g)
apple_pi
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:D
apple_pi
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I see one problem though... this will only be accurate with small angles right?