No Title

Instructions:

This is an open book, open note takehome exam, due Monday 2/25 by 5 pm. Feel free to ask me any clarifying questions on the problems, but do not consult otherwise with anyone on this exam. This exam is subject to the Emory Honor Code.

Summary:

Your NAME________________________

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Your SCORE____

A projectile is shot upward with initial velocity and is subject both to gravity and to a drag force of magnitude , where c is constant.
1. Using direct integration techniques on Newton's second law, solve for v(x), the dependence of the particle's velocity on its height x.
2. To what maximum height h does the projectile reach?
3. From homework 1, assigned problem 2, we know that the projectile as it returns to earth obeys
  
  where the terminal velocity .
  With what velocity does the projectile strike the ground? (Express your answer in terms of and , the velocity the particle was initially shot up with in part (a).) Why does this make physical sense, from an energetics point of view?
A particle of mass m moves in a conservative force field described by the potential energy

where a and c are positive constants.
1. What are the equilibrium positions for the particle?
2. Which are stable?
3. Give the frequency of oscillation about equilibrium, for any stable equilibrium points.
4. Sketch V(x) qualitatively.
5. For what range of total energy E does the particle undergo bound oscillations?
1. What is the equation of motion for the mass m?
2. What is the homogeneous solution?
3. What is the particular solution?
4. What is the motion x(t) of the mass m, if it begins at rest a distance beneath the table?
5. If we wait a very long time, what motion x(t) will we observe for the mass m?
Consider an LR circuit whose voltage source carries voltage (that is, B & O Figure 1-9 with C=0 and ).
1. Write the differential equation for the current i(t). What is the general solution for the homogeneous problem, for this ODE?
2. Using the approach of B & O section 1-9, find a particular solution of the form
  
  That is, find and . Plot the phase lag .
3. If the circuit initially has zero current, find the current i(t).
In analyzing LRC circuits, we introduce a complex impedance, which relates an applied sinusoidal voltage to the induced current in a generalization of Ohm's law:

Because Z is complex, it describes both the amplitude and phase lag of the response current.
1. Consider a circuit containing only a voltage source and a capacitor. By studying the particular solution to the Kirchoff's law ODE, derive the expression for capacitive impedance
2. Similarly, consider circuits containing first, only a voltage source and a resistor; and second, only a voltage source and an inductor. Treating Kirchoff's law as an ODE for the current i(t), derive expressions for the impedances and of a resistor and an inductor, respectively.
3. For a series LRC circuit, where the current through each circuit element is instantaneously the same, explain why your values for the impedances of R, L, and C agree with the conventional phasor diagram below, which shows voltage across a resistor being in phase with the current, while voltage across the inductor leads the current by a phase angle, and voltage across the capacitor lags the current by .
4. For the LR circuit of problem 4, note that is just the impedance Z of the circuit. Show that this impedance is just the sum .
5. For the LR circuit of problem 4, in the special case where , plot a phasor diagram like that of part (c), showing the phase relationships between the applied voltage , and the steady state: current I induced in the circuit, voltage across the resistor, and voltage across the inductor.