Results 1 to 3 of 3

Math Help - Simple Harmonic Motion

  1. #1
    Junior Member
    Joined
    Nov 2009
    Posts
    60

    Simple Harmonic Motion

    Regarding Simple Harmonic Motion, and a spring moving up and down, when the displacement is largest apparently the acceleration is also maximum. But WHY? When the distance is biggest, isnt speed 0 so the system is also not accelerating?!

    So why is acceleration maximum at maximum displacement?!

    help much appreciated! thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello Yehia
    Quote Originally Posted by Yehia View Post
    Regarding Simple Harmonic Motion, and a spring moving up and down, when the displacement is largest apparently the acceleration is also maximum.
    You are quite right: when the displacement is largest, the acceleration is also a maximum.
    But WHY?
    Because this is the basic characteristic of SHM - the acceleration is proportional to the displacement. In mathematical terms, this is represented by the fundamental equation of SHM:
    \underbrace{\ddot{x}}_{\text{acceleration }} \underbrace{= -\omega^2}_{\text{ is proportional to }}\underbrace{x}_{\text{ displacement}}
    The minus sign in this equation means that the acceleration is always in the opposite direction to the displacement. In other words, the acceleration is always directed towards the centre of the motion.
    When the distance is biggest, isnt speed 0 so the system is also not accelerating?!
    Not at all! You're confusing speed with acceleration. It is perfectly possible for a body to have zero speed and yet be accelerating. This is exactly what happens at the extremities of the simple harmonic motion.

    If you want another illustration of zero speed, but non-zero acceleration, consider a ball that is thrown vertically upwards. As you know, it is accelerating downwards with a constant gravitational acceleration g. During the upward phase, gravity is slowing the ball down. At the top of its flight, its velocity is zero; then gravity starts to increase the speed for the downward part of the motion. At the top, the speed is zero, but the acceleration is a constant g downwards for the whole flight!

    Grandad
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2009
    Posts
    60
    Quote Originally Posted by Grandad View Post
    Hello YehiaYou are quite right: when the displacement is largest, the acceleration is also a maximum. Because this is the basic characteristic of SHM - the acceleration is proportional to the displacement. In mathematical terms, this is represented by the fundamental equation of SHM:
    \underbrace{\ddot{x}}_{\text{acceleration }} \underbrace{= -\omega^2}_{\text{ is proportional to }}\underbrace{x}_{\text{ displacement}}
    The minus sign in this equation means that the acceleration is always in the opposite direction to the displacement. In other words, the acceleration is always directed towards the centre of the motion.Not at all! You're confusing speed with acceleration. It is perfectly possible for a body to have zero speed and yet be accelerating. This is exactly what happens at the extremities of the simple harmonic motion.

    If you want another illustration of zero speed, but non-zero acceleration, consider a ball that is thrown vertically upwards. As you know, it is accelerating downwards with a constant gravitational acceleration g. During the upward phase, gravity is slowing the ball down. At the top of its flight, its velocity is zero; then gravity starts to increase the speed for the downward part of the motion. At the top, the speed is zero, but the acceleration is a constant g downwards for the whole flight!

    Grandad
    Right! I see it now. as you said, i confused Velocity with acceleration. Thanks a lot for clarifying this for me! fingers crossed on my physics exam next week now...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simple Harmonic Motion 2
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 8th 2010, 08:55 AM
  2. Simple Harmonic Motion
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: May 16th 2009, 06:14 AM
  3. simple harmonic motion
    Posted in the Advanced Applied Math Forum
    Replies: 10
    Last Post: January 4th 2008, 01:09 PM
  4. Simple Harmonic Motion
    Posted in the Trigonometry Forum
    Replies: 8
    Last Post: October 11th 2007, 11:22 AM
  5. Simple Harmonic Motion
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: August 25th 2006, 02:23 PM

Search Tags


/mathhelpforum @mathhelpforum