13: Equation of Motion for Simple Harmonic Oscillators

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Topics covered: The conservation of mechanical energy can be used to derive the equation of motion for simple harmonic oscillators (SHO).

Instructor/speaker: Prof. Walter Lewin

Date recorded: October 8, 1999

Video Index

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  • Gravitational Potential Energy
    A review is given of the spatial dependence of the gravitational potential energy both close to the Earth's surface and at large distances from Earth. The gravitational force pulls objects in the direction of decreasing potential energy.

  • Calculating U(x) from F(x) and Vice Versa
    The potential energy, U(x), of a spring system is derived and sketched as a function of displacement x. The force can be derived if the function U(x) is known: F(x)=-dU(x)/dx.

  • Equilibrium Points
    The minima and maxima of potential energy are positions where the net force is zero. At the stable equilibrium points the 2nd derivative of U(x) is positive, at the unstable equilibrium points the 2nd derivative is negative.

  • Parabolic Potential Energy Well ==> SHO
    Using the parabolic shape of the potential energy for a spring, and the conservation of mechanical energy, it is shown that the mass on the spring oscillates as a simple harmonic oscillator (SHO).

  • Circular Potential Energy Well ==> SHO
    Using a circular potential energy well and the conservation of mechanical energy, it is shown that for SMALL ANGLES, the oscillations are simple harmonic. A circular track with very large radius is used to demonstrate this.

  • Sliding on a Circular Track ==> SHO
    The known radius of a circular air track is used to predict the period of oscillation of a sliding object (small angles!), and a measurement is made to confirm this. The process is repeated for a ball bearing rolling in another circular track. The period of oscillation can now not be predicted in a similar way as was possible in the case of the air track. Why? ==> No, it has nothing to do with friction!

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