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Differential equation for a pendulum

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  1. Apr 12, 2017 #1
    1. The problem statement, all variables and given/known data
    A simple pendulum is formed by a light string of length ##l## and with a small bob ##B## of mass ##m## at one end. The strings hang from a fixed point at another end. The string makes an angle ##\theta## with the vertical at time ##t##. Write down an equation of motion of ##B## perpendicular to ##OB## and state the approximation that enables you to treat this equation as an SHM equation. State the general solution of this equation.

    3. The attempt at a solution
    Equation of motion : ## \rm \large -mgl\sin(\pi -\theta) = ml^2 \ddot \theta##
    Rearranges to: ## \rm \large \ddot \theta =-\frac{g}{l}sin(\pi-\theta)##
    Approximation: ## \rm \large sin \theta \approx \theta## provided ## \theta ## is small.
    General solution: ## \rm \large \theta = \theta_0sin(\sqrt{\frac{g}{l}}t)##

    Are my answers correct?
    The reason I am confused is because I think I have misunderstood what they mean by "equation of motion of ##B## perpendicular to ##OB##" and may have got all of my answers wrong.
    And secondly I am well aware of finding a general solution of a differential equation of the form ## \rm \small \ddot y =ay ##, however, I have never seen a general solution of a differential equation of the form ##
    \rm \small \ddot y =a(\lambda-y) ## where ##\lambda## is a constant.
     
    Last edited: Apr 12, 2017
  2. jcsd
  3. Apr 12, 2017 #2
    It is not entirely clear what you are calling theta (and thus why you have involved a pi - theta argument for the sine). A figure would really help your question, along with labels on the figure.

    As to the solution for the differential equation, did you really mean ddy = a*y, where a is presumed to be positive? Or did you intend to have it be negative?

    The additional term a*lambda simply adds a constant to the particular solution.
     
  4. Apr 12, 2017 #3
    Why are you interested in the solution to that differential equation?
     
  5. Apr 12, 2017 #4
    WhatsApp Image 2017-04-13 at 5.22.17 AM.jpeg
    I hope this will work.

    Yes, can you show me how to add the constant to the particular solution? Like for example if the solution to the differential equation ## \ddot y = ay## where a is a constant is ## y = y_0 \sin \omega t##, how will the constant term be added in the solution if the equation is changed to ## \ddot y = a(\lambda-y)##? (If changes vary from equation to equation, please refer to the SHM equation given in my first post) If you find my example vague, feel free to make one of yours.
    I am not interested in that particular equation. I am just concerned whether changes will occur in the solution if the concerned variable was increased or decreased by a constant term.
     
  6. Apr 12, 2017 #5
    Start by replacing sin(pi-theta) with just sin(theta); they are exactly equal.

    Then, with a little bit of re-arrangement, your equation should look like
    dd(theta) + w^2*sin(theta) = 0

    Then make the approximation that for small theta, sin(theta) = theta, so you have
    dd(theta) + w^2*theta = 0

    That should be the only equation you need to solve for this problem.
    I really don't know where the question about a constant term on the right came from.
     
  7. Apr 13, 2017 #6
    Okay thank you that explains a lot.
     
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