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apple_pi

  • 3 years ago

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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  1. akash123
    • 3 years ago
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    write down the equation of motion for pendulum

  2. akash123
    • 3 years ago
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    |dw:1349777134567:dw|

  3. apple_pi
    • 3 years ago
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    why is the mg force not straight down?

  4. akash123
    • 3 years ago
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    sorry...mg is straight down only

  5. akash123
    • 3 years ago
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    does it help u in finding the time period?

  6. apple_pi
    • 3 years ago
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    I want to know how the formula is derived...

  7. akash123
    • 3 years ago
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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

  8. akash123
    • 3 years ago
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    it's fine?

  9. akash123
    • 3 years ago
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    net force along tangent is mg sin (theta)

  10. apple_pi
    • 3 years ago
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    yes

  11. akash123
    • 3 years ago
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    now v_tangential= r ( angular vel)

  12. akash123
    • 3 years ago
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    differentiate it a_tangential= r(angular acc)

  13. apple_pi
    • 3 years ago
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    ok

  14. akash123
    • 3 years ago
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    r is the length of pendulum in this case

  15. akash123
    • 3 years ago
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    now substitute a_tangential into m( a_t) = mg sin(theta)

  16. apple_pi
    • 3 years ago
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    m(r*angular_acc) = m(sin(theta))

  17. akash123
    • 3 years ago
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    m ( l(angular acc))= mg sin(theta)....l=r= length of pendulum

  18. akash123
    • 3 years ago
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    now l(angular acc))= g sin(theta)

  19. apple_pi
    • 3 years ago
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    where did the l come from?

  20. akash123
    • 3 years ago
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    l=r=length of pendulum...pendulum is like bob is moving in a circle pivoted at centre

  21. apple_pi
    • 3 years ago
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    oh right... l = r. I forgot about that

  22. akash123
    • 3 years ago
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    now acceleration = g/l sin(theta)

  23. apple_pi
    • 3 years ago
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    yep

  24. akash123
    • 3 years ago
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    equation of SHM is a= - w^2 x then time period T= 2 pi/w

  25. apple_pi
    • 3 years ago
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    I havent learnt SHM yet. whats w?

  26. akash123
    • 3 years ago
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    w(omega)---angular frequency acceleration = g/l sin(theta) for small theta acceleration = g/l (theta) sin(theta)= theta

  27. akash123
    • 3 years ago
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    u know wave... omega= 2 pi (nu)

  28. apple_pi
    • 3 years ago
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    yep

  29. akash123
    • 3 years ago
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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.

  30. apple_pi
    • 3 years ago
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    ok. I know the basics of it

  31. akash123
    • 3 years ago
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    ok.. so acceleration = g/l (theta)

  32. akash123
    • 3 years ago
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    for small theta

  33. apple_pi
    • 3 years ago
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    yep

  34. akash123
    • 3 years ago
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    now compare this with a= - w^2 x where x is the displacement from fixed point

  35. apple_pi
    • 3 years ago
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    so for small angle x = theta

  36. akash123
    • 3 years ago
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    so acceleration and displacement are in opposite direction

  37. apple_pi
    • 3 years ago
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    yes seems logical

  38. akash123
    • 3 years ago
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    here theta is x.... for small angle sin(theta)= theta

  39. akash123
    • 3 years ago
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    so in the case of pendulum angular displacement and angular acceleration are in opposite direction

  40. akash123
    • 3 years ago
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    write six in the form a power series sin x= x - x^3/3! + x^5/5!-------

  41. akash123
    • 3 years ago
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    so for small x sin x =x

  42. akash123
    • 3 years ago
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    neglect higher order terms

  43. apple_pi
    • 3 years ago
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    aha

  44. akash123
    • 3 years ago
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    so we have acceleration =- g/l (theta)

  45. apple_pi
    • 3 years ago
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    yes

  46. akash123
    • 3 years ago
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    so u have w^2= g/l

  47. akash123
    • 3 years ago
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    so T= 2 pi/w

  48. apple_pi
    • 3 years ago
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    I forgot what w is equal to

  49. akash123
    • 3 years ago
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    w= angular frequency= 2 pi( frequency)

  50. apple_pi
    • 3 years ago
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    so T = 2pi/2pi?

  51. akash123
    • 3 years ago
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    T= (2 pi)/ (2 pi( frequency))

  52. akash123
    • 3 years ago
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    :)

  53. akash123
    • 3 years ago
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    so T= 1/ (frequency)

  54. apple_pi
    • 3 years ago
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    so where does the √ come in?

  55. akash123
    • 3 years ago
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    w^2= g/l so what will be w?

  56. apple_pi
    • 3 years ago
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    √(g/l)

  57. akash123
    • 3 years ago
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    so T=?

  58. apple_pi
    • 3 years ago
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    2π T=----- √(g/l)

  59. akash123
    • 3 years ago
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    so u got the time period.

  60. apple_pi
    • 3 years ago
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    =2π√(l/g)

  61. apple_pi
    • 3 years ago
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    :D

  62. apple_pi
    • 3 years ago
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    I see one problem though... this will only be accurate with small angles right?

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