Midterm Exam Fall 01 PH 314 Laundry List

• solve for x(t), v(t) given f(x)
• solve for x(t), v(t) given f(v)
• solve for x(v) given f(v)

• simple harmonic oscillator (HO):
• ma = -k x => x = A exp (i omega t) = A cos (omega t -delta)
• show omega^2 = k/m, take omega_o as sqrt(k/m) for undamped HO

• know relation of force and potential ( relation involves the gradient )
• taylor expansion of a function f(x) = f(a) + (x-a)f'(a) + ...
• find the frequency of small oscillations around the minimum in a given potential V(x)

• damped oscillator: Equation of motion is m a = - b x-dot - k x = -2 m beta x-dot - k x
• let x(t) = A exp( alpha t) and show that alpha = - beta plus/minus sqrt(beta^2 - omega_o^2)
• 3 cases: underdamped, critically damped, overdamped oscillations
• underdamped oscillation:
• x(t) = A exp(-beta t) exp( i omega t)        (how is omega related to omega_o ?)
• Q = energy/(energy loss per radian)
• show that Q = | omega E / (power loss) |
• show that d<E>/dt = - 2 beta <E>, so <E> = <Eo> exp(- 2 beta t)
• Be able to find Q from a rate of energy loss
• Identify the form of x(t) for critically damped motion
• driven, underdamped oscillator
• Fo exp( i omega t) -2 m beta x-dot - k x = ma. Let x(t) = D exp(i omega t)
• Show that D = Do exp(-i delta), where
• Do = |D| = Fo/m /sqrt( (omega_o^2-omega^2)^2 + (2 omega beta)^2) and
• tan (delta) = 2 omega beta/(omega_o^2-omega^2)
• Find Q from omega_r/(delta-omega), where delta-omega is the 'width' between the half-power points
• Identify each element in a series R-L-C circuit driven by sinusoidal emf Eo with an element in the mechanical system. Start by writing down the correct loop equation for the RLC circuit.
• Coupled oscillations of two masses and 2 or 3 springs
• obtain the frequencies of oscillation
• find the amplitude ratios for each frequency
• (I won't ask you to find Q1 and Q2, the 'normal mode' combinations of coordinates a x1 + b x2)