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Homework Help: Pendulum with horizontal spring

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data

    A 5kg sphere is connected to a thin massless but rigid rod of length L=1.3 m to form a simple pendulum. The rod is connected to a nearby vertical wall by a spring with a spring constant k= 75N/m, connected to it at a distance h=1.1 m below the pivot point of the pendulum. What is the angular frequency (in rad/s) of the system for small amplitude oscillations.



    2. Relevant equations

    [tex]\alpha[/tex] = [tex]\omega[/tex]2*[tex]\theta[/tex]
    I=m*r2
    Torque=f*d*sin[tex]\theta[/tex]

    or small oscillations, sin[tex]\theta[/tex]=[tex]\theta[/tex] and cos[tex]\theta[/tex]=1



    3. The attempt at a solution

    sum the torque and set them equal to [tex]\alpha[/tex]I

    gravitational torque= Lmg[tex]\theta[/tex]

    if theta is the angle between the rod and its equilibrium position, the angle of the spring torque is (theta + [tex]\pi[/tex]/2), so the spring torque is
    f*d*sin(theta + [tex]\pi[/tex]/2). The force is k*[tex]\Delta[/tex]x, or h*sin(theta + [tex]\pi[/tex]/2) and the distance is h, giving

    torque from spring =k*h2*sin[tex]\theta[/tex]*sin(theta + [tex]\pi[/tex]/2)
    or k*h2*[tex]\theta[/tex]*(theta + [tex]\pi[/tex]/2)

    My final equation is

    mL2[tex]\omega[/tex]2tex]\theta[/tex] = Lmg[tex]\theta[/tex] + k*h2*[tex]\theta[/tex]*(theta + [tex]\pi[/tex]/2)

    The problem is that this gives me a [tex]\omega[/tex] that is dependant on [tex]\theta[/tex], which isn't possible. It should be constant. Can someone please tell me where I'm going wrong?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Apr 27, 2010 #2

    ehild

    User Avatar
    Homework Helper

    sin(theta+pi/2)= cos(theta). For small angles, cos(theta) =1.

    ehild
     
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