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    Physical Pendulum: Rotational Simple Harmonic Motion.

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    Question: A straight uniform stick having a length b (m) and a mass M (kg) is freely pivoted at one end. Find the frequency of oscillation about the pivot, assuming the angle theta is small.

    © BrainMass Inc. brainmass.com December 24, 2021, 6:56 pm ad1c9bdddf
    https://brainmass.com/physics/torques/physical-pendulum-rotational-simple-harmonic-motion-151553

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    The problem can be solved easily by considering rotational simple harmonic motion.

    For a linear simple harmonic motion force acting on a particle is directly proportional to the displacement of the particle from equilibrium position and directed to the equilibrium position and hence the equation of motion is given by

    F = - K*x

    Where F is the force and K is the constant may be called the force constant.

    The frequency of oscillation is given by

    Where m is the mass of the particle

    In the same way a body is called in rotational SHM for small angles if the torque acting on it is directly proportional to the angular displacement  and given by

     = - K*

    and the frequency of small oscillation is given analogues to the above equation as

    ---------------------------- (1)

    Where I is the moment of inertia of a body

    Now if the rod is slightly displaced from its equilibrium position, the only restoring torque acting on it is due to its weight Mg and is given by

    ---------------------------- (2)

    Negative sign due to restoring torque

    If  is small enough and measured in radians we know that sin  =  and hence we have

    ---------------------------- (2)

    And hence the value of K for the rotational SHM can be given by

    ---------------------------- (3)

    The moment of the inertia of the stick about one end is given by

    ---------------------------- (4)

    Hence substituting values from equations 4 and 3 in 1 we get the frequency of small oscillations of the rod as

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 6:56 pm ad1c9bdddf>
    https://brainmass.com/physics/torques/physical-pendulum-rotational-simple-harmonic-motion-151553

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