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Pendulum velocity at any given angle

  1. Feb 10, 2014 #1
    Trying to find an equation that will give me a pendulum bob velocity at any given angle if the pendulum is released from a 90° angle from it's equilibrium position. My parameters are as follows

    9.8 (acceleration of gravity m/s^2)
    1.397 (length of pendulum in meters)
    .138 (mass of pendulum bob in kilograms)
    90 (Θmax, maximum angle of the pendulum in degrees)
    45 (Θ, the angle of interest for velocity in degrees)

    Below is the equation that I found

    v=√2*g*l/m*(cos(Θ) – cos(Θmax))

    My confusion is that the equation for pendulum velocity at it's equilibrium position which should be the maximum velocity is less that at 45° in the above equation


    I understand that mass is in the first equation and not the second equation, but is the first equation correct for velocity at any angle? Is the equation for maximum velocity at the pendulum equilibrium position not correct? Any help would be appreciated.
  2. jcsd
  3. Feb 10, 2014 #2


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    I don't know where your equations are coming from. The speed at any point is independent of the mass. You are correct that the max speed is at the bottom of the swing and equal to v=√2*g*l = 5.23 m/s which comes from conservation of energy ...initial potential energy = final kinetic energy......you can do the same at 45 degrees to calculate the speed, which must be less than the max.....the initial PE is mgh where h from some trig is 1.397sin45.....then mgh = 1/2mv^2, m cancels, solve for v = 4.4 m/s.
  4. Feb 10, 2014 #3
    cos(Θmax) is zero (the maximum angle is 90 degrees, isn't it?)
    So your formula should be

    which is maximum for Θ=0.

    Cos of 45 is √2/2.
  5. Feb 10, 2014 #4
    There should be no mass in that equation.
  6. Feb 10, 2014 #5
    Thanks for the feedback, ended up answering my own question. Went at it from the energy conservation standpoint.

    a=[90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0]; // angle to calculate velocity at

    g=9.8; //acceleration of gravity (meters/sec^2)
    l=1.397; // length (meters)
    ao=90; // maximum angle (degrees)
    m = .138; // mass (kilograms)

    ME=m*g*l // ME = PE + KE, PE=ME at 90 degree max bob height, KE=0

    h=l-(cos(a)*l) // pendulum bob height at a given angle
    PE=m*g*h // Potential Energy calculation
    KE=ME-PE // Kinetic energy at a given angle
    v_angle=sqrt(KE*2/m) // pendulum bob velocity at a given angle

    plot of angle vs velocity (see attached JPG file)

    At 90° (max pendulum bob height) 0 meters/sec velocity
    AT 0° (0 pendulum bob height) equilibrium position 5.23 meters/sec velocity

    Attached Files:

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