Help   |   Contact Us

 

Lecture 2: Damped Free Oscillations

OCW Scholar

« Previous | Next »

Lecture Topics

An orange disc sits within a larger grey disc.
  • Beats
  • Damped Free Oscillations (Under-, Over-, and Critically Damped)
  • Quality Factor

Learning Objectives

By the end of this lecture, you should:

  • understand how to combine (superpose) two vibrations and show that beats arise.
  • know what beats sound like and how they look on an oscilloscope or graph.
  • know how to use a viscous frictional force in Newton's second law to solve for damped oscillatory motion.
  • know that a viscous force causes an exponential decay and a slightly lower oscillation frequency.
  • understand the quality factor.
  • draw a simple RLC circuit and understand the physics behind its components.
  • know the solution of an RLC circuit without necessarily understanding the derivation.
  • understand underdamping, overdamping, and critical damping as behaviors (draw graphs).

Lecture Activities

Check Yourself

  • A typical small inductor is a little coil of value 22 μH. Used in an RLC circuit with a typical small capacitor of 10 pF, what would be the frequency (in Hz)?

› View/hide answer

    10.7 MHz, which happens to be a common frequency in the internal circuitry of FM radios.

 

  • It turns out that the small inductor referred to above has a resistance of 1.1 Ω, while resistance due to the capacitor need not be considered in the circuit. What would be the quality factor of the circuit made with these real parts?

     

› View/hide answer

    About 1300, making a relatively narrow resonance peak as expected for an electronic circuit.

 

  • When an oscillator is not driven, although it may have an initial velocity and displacement, and in the special case that the damping rate γ is twice the resonant frequency, the critical damping curve describes the motion. Give details, including the form of the curve

     

› View/hide answer

    The curve is dominated by exponential decay to the equilibrium position, but initially has a displacement which can even increase if there is an initial velocity. The curve plays off a linear function vs. the exponential decay:

    \[x=(A+Bt){{e}^{-\frac{\gamma }{2}t}}\]

 

  • If \({{\omega }_{0}}\)is the resonant angular frequency of a system, you might expect that the frequency would be less with damping, and how much less would depend on the amount of damping. Write down the actual frequency for a damped but undriven oscillator.

     

› View/hide answer

    \[\omega =\sqrt{\omega _{0}^{2}-\frac{{{\gamma }^{2}}}{4}}\] which is a positive number and less than the undamped value.

 

  • A tuning fork of 500 Hz has its amplitude of vibration decrease in one minute to 5% of its original value when first struck. What is the quality factor Q?

     

› View/hide answer

    31415 (about 10000π). Note that \({{e}^{-3}}\approx 0.05\).

 

« Previous | Next »