Results 1 to 2 of 2

Math Help - [SOLVED] Time elapsed (SHM)?

  1. #1
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
    Posts
    539

    [SOLVED] Time elapsed (SHM)?

    A particle executes SHM with period T about a point O. It passes through a point P with velocity V along OP. Show that the time that elapses when it again comes to P is given by: t = \frac{T}{\pi}\cdot \arctan \left(\frac{TV}{2\pi\cdot OP}\right)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    May 2009
    Posts
    127
    Here's how to do it:

    Spoiler:

    It is well known that the solution for SHM with ( x=0 when t=0) is:

    x = A \sin (\omega t)

    with

    \dot{x} = A \omega \cos (\omega t)

    where A is the amplitude of oscillations and \omega = \frac{2 \pi}{T}.

    Now in the first instance when the particle is at P let the time be t_0 and let d = |\vec{\mathbf{OP}}| so then

    d = A \sin (\omega t_0 ) and V = A \omega \cos (\omega t_0)

    which after dividing gives us:

    \boxed{\frac{\omega d}{V} = \tan (\omega t_0 )} .

    Similarly in the second instance when the particle is at P let the time be t_1 and so then

    d = A \sin (\omega t_1 ) and -V = A \omega \cos (\omega t_1)

    which after dividing gives us:

    \boxed{-\frac{\omega d}{V} = \tan (\omega t_1 )}. (Note this is in 2nd quadrant).

    Thus (taking into account the quadrants) we have:

    \omega (t_1 - t_0 ) = \arctan \left( \frac{\omega d}{-V} \right) - \arctan \left( \frac{\omega d}{V} \right)
    = \pi - 2 \arctan \left( \frac{\omega d}{V} \right)

    so

    \frac{\omega d}{V} = \tan \left( \frac{\pi}{2} - \frac{\omega}{2} (t_1 - t_0 ) \right)

    = \cot \left( \frac{\omega}{2} (t_1 - t_0 ) \right)

    = \cot \left( \frac{\pi}{T} (t_1 - t_0 ) \right)

    \Leftrightarrow \tan \left( \frac{\pi}{T} (t_1 - t_0 ) \right) = \frac{V}{\omega d}

    Hence

    {\color[rgb]{0,0,1} \boxed{t = t_1 - t_0 = \frac{T}{\pi} \arctan \left( \frac{TV}{2 \pi d} \right)}} .

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. how many hours elapsed when the body was found...
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: February 21st 2011, 05:39 PM
  2. Replies: 2
    Last Post: March 24th 2010, 08:22 PM
  3. [SOLVED] Polynomial time/height
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: January 27th 2010, 09:48 AM
  4. [SOLVED] Time period of oscillations?
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: May 30th 2009, 12:48 AM
  5. Replies: 8
    Last Post: March 3rd 2009, 11:14 AM

Search Tags


/mathhelpforum @mathhelpforum