From b61c3cac8bd1f3a71d28dcf90e84c0d827666db0 Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Tue, 30 Jun 2009 16:33:37 -0400 Subject: [PATCH] Working on PHYS 102 recitation 1 solutions --- asymptote/ElectroMag.asy | 104 +++--------------- .../Serway_and_Jewett_4/problem19.07.tex | 70 +++++++----- .../Young_and_Freedman_12/problem21.86.tex | 28 +++++ 3 files changed, 88 insertions(+), 114 deletions(-) diff --git a/asymptote/ElectroMag.asy b/asymptote/ElectroMag.asy index 7db486c..4754b15 100644 --- a/asymptote/ElectroMag.asy +++ b/asymptote/ElectroMag.asy @@ -3,6 +3,7 @@ import Mechanics; // ---------------------- Charges ------------------------- +// charged particle struct Charge { pair center; real q; @@ -29,46 +30,49 @@ struct Charge { } } +// positive charge Charge pCharge(pair center=(0,0), real q=1, real radius=2mm, pen outline=currentpen, Label L="") { Charge c = Charge(center=center, q=q, radius=radius, outline=outline, L=L, fill=red); return c; } +// negative charge Charge nCharge(pair center=(0,0), real q=-1, real radius=2mm, pen outline=currentpen, Label L="") { Charge c = Charge(center=center, q=q, radius=radius, outline=outline, L=L, fill=blue); return c; } +// auto-signed charge +Charge aCharge(pair center=(0,0), real q=1, real radius=2mm, pen outline=currentpen, Label L="") +{ + if (q > 0) { + Charge c = pCharge(center, q, radius, outline, L); + } else { + Charge c = nCharge(center, q, radius, outline, L); + } + return c; +} + // ---------------------- Vectors ------------------------- +// electric field Vector EField(pair center=(0,0), real mag=5mm, real dir=0, Label L="") { Vector v = Vector(center=center, mag=mag, dir=dir, L=L, outline=rgb(1,0.5,0.2)); // orange return v; } +// magnetic field Vector BField(pair center=(0,0), real mag=5mm, real dir=0, Label L="") { Vector v = Vector(center=center, mag=mag, dir=dir, L=L, outline=rgb(0.1,1,0.2)); // green return v; } -Vector Velocity(pair center=(0,0), real mag=5mm, real dir=0, Label L="") -{ - Vector v = Vector(center=center, mag=mag, dir=dir, L=L, outline=rgb(1,0.1,0.2)); // red - return v; -} - // ---------------------- Forces ------------------------- -Vector Force(pair center=(0,0), real mag=5mm, real dir=0, Label L="") -{ - Vector v = Vector(center=center, mag=mag, dir=dir, L=L, outline=rgb(0.1,0.2,1)); // blue - return v; -} - // Force of a on b Vector CoulombForce(Charge a, Charge b, Label L="", real scale=1mm, real unit=1mm) { @@ -94,80 +98,4 @@ void CoulombForces(Charge c[], real scale=1mm, real unit=1mm) } } -// ---------------------- Measures ------------------------- - -// Distance derived from CAD.MeasuredLine -struct Distance { - pair pFrom; - pair pTo; - real scale; - pen outline; - Label L; - - void operator init(pair pFrom=(0,0), pair pTo=(5mm,0), real scale=5mm, pen outline=currentpen, Label L="") { - this.pFrom = pFrom; - this.pTo = pTo; - this.outline = outline; - this.L = L; - } - - void draw(picture pic=currentpicture) { - picture picF; - picture picL; - label(picL, L); - pair pLabelSize = 1.2 * (max(picL)-min(picL)); - pair pDiff = pTo - pFrom; - path p = (0,0)--pDiff; - draw(picF, p, outline, Arrows); - label(pic = picF, - L = rotate(degrees(pDiff)) * L, - position = - pDiff/2 - + unit(rotate(90)*pDiff) * pLabelSize.y / 2); - add(pic, picF, pFrom); - } -} - -struct Angle { - pair B; - pair A; - pair C; - real radius; // radius < 0 for exterior angles. - pen outline; - Label L; - - void operator init(pair B, pair A, pair C, real radius=5mm, pen outline=currentpen, Label L="") { - this.B = B; - this.A = A; - this.C = C; - this.radius = radius; - this.outline = outline; - this.L = L; - } - - void draw(picture pic=currentpicture) { - picture picF; - picture picL; - label(picL, L); - pair pLabelSize = 1.2 * (max(picL)-min(picL)); - path p = arc(B-A, (0,0), C-A, radius); - real t = reltime(p, 0.5); - pair P = midpoint(p); - pair tang = dir(p, t); - - draw(picF, p, outline); - label(pic = picF, - L = rotate(tang) * L, - position = - P + unit(P) * pLabelSize.y / 2); - add(pic, picF, A); - } -} - -// TODO: ihat, ijhat - -// ---------------------- Shapes ------------------------- - -// TODO: ring, plate, block, cylinder, spring, table - diff --git a/latex/problems/Serway_and_Jewett_4/problem19.07.tex b/latex/problems/Serway_and_Jewett_4/problem19.07.tex index fdfcf4d..1e1db2f 100644 --- a/latex/problems/Serway_and_Jewett_4/problem19.07.tex +++ b/latex/problems/Serway_and_Jewett_4/problem19.07.tex @@ -7,25 +7,33 @@ and the other a charge of $q_2 = -18.0\U{nC}$. Find the electric force between the two after they have come to equilibrium. \end{problem*} % problem 19.7 -\empaddtoprelude{ - pair A, B; - A := origin; - B := (3cm, 0); -} +% A := origin; +% B := (3cm, 0); \begin{solution} \Part{a} \begin{center} -\begin{empfile}[1a] -\begin{emp}(0cm, 0cm) - label.top("F", draw_force(A, B, -30pt)); - draw_pcharge(A, 5pt); - label.llft(btex $q_1$ etex, A+6pt*dir(-135)); - draw_ncharge(B, 6pt); - label.lrt(btex $q_2$ etex, B+6pt*dir(-45)); - label.bot("r", draw_length(A, B, 10pt)); -\end{emp} -\end{empfile} +\begin{asy} +import Mechanics; +import ElectroMag; + +real u = 1cm; // Length of 1 m on the page +real Fscale = .3cm; // Length of 1 N on the page +qa = 12e-9; +qb = -18e-9 + +Charge a = aCharge((0,0)*u, q=qa, L="$q_1$"); +Charge b = aCharge((0,0.03)*u, q=qb, L="$q_2$"); +Distance r = Distance(a.center, b.center, L="$r$"); +Vector Fab = CoulombForce(a, b, scale=Fscale, L="$F$"); +Vector Fba = CoulombForce(b, a, scale=Fscale, L="$F$"); + +r.draw(); +Fab.draw(); +Fba.draw(); +a.draw(); +b.draw(); +\end{asy} \end{center} \begin{equation} F = k_e \frac{q_1 q_2}{r^2} @@ -37,17 +45,27 @@ opposites attract. \Part{b} \begin{center} -\begin{empfile}[1b] -\begin{emp}(0cm, 0cm) - label.top("F", draw_force(A, B, 10pt)); - draw A--B withcolor (.7,.7,.7) withpen pencircle scaled 1pt; - draw_ncharge(A, 3pt); - label.llft(btex $Q/2$ etex, A+6pt*dir(-135)); - draw_ncharge(B, 3pt); - label.lrt(btex $Q/2$ etex, B+6pt*dir(-45)); - label.bot("r", draw_length(A, B, 10pt)); -\end{emp} -\end{empfile} +\begin{asy} +import Mechanics; +import ElectroMag; + +real u = 1cm; // Length of 1 m on the page +real Fscale = .3cm; // Length of 1 N on the page +qa = 12e-9; +qb = -18e-9 + +Charge a = aCharge((0,0)*u, q=(qa+qb)/2, L="$Q/2$"); +Charge b = aCharge((0,0.03)*u, q=(qa+qb)/2, L="$Q/2$"); +Distance r = Distance(a.center, b.center, L="$r$"); +Vector Fab = CoulombForce(a, b, scale=Fscale, L="$F$"); +Vector Fba = CoulombForce(b, a, scale=Fscale, L="$F$"); + +r.draw(); +Fab.draw(); +Fba.draw(); +a.draw(); +b.draw(); +\end{asy} \end{center} The total charge on the both spheres is $Q = q_1 + q_2 = -6.0\U{nC}$. The spheres are identical, so at equilibrium, there will be $Q/2 = diff --git a/latex/problems/Young_and_Freedman_12/problem21.86.tex b/latex/problems/Young_and_Freedman_12/problem21.86.tex index 8a2cbd0..35cbdfc 100644 --- a/latex/problems/Young_and_Freedman_12/problem21.86.tex +++ b/latex/problems/Young_and_Freedman_12/problem21.86.tex @@ -12,4 +12,32 @@ charge must be given to the drop? \end{problem*} \begin{solution} +\begin{center} +\begin{asy} + +\end{asy} +\end{center} +From the forces on the drop in each direction +\begin{align} + F_x &= 0 = m a_x & + F_y &= qE = m a_y \\ + a_x &= 0 & + a_y &= \frac{qE}{m} \;. +\end{align} +Now this looks like a projectile motion problem from your +intro-mechanics class. No acceleration in the $x$ direction means +$v_x$ is constant, so the time-of-flight is given by. +\begin{align} + \Delta x &= v_x \Delta t & + \Delta t &= \frac{\Delta x}{v_x} = 1\U{ms} \;. +\end{align} +We can plug this time-of-flight into our constant-acceleration +equation for $y(t)$, +\begin{align} + y(t) &= \frac{a_y}{2}t^2 + v_{y0} t + y_0 \\ + \Delta y &= \frac{qE}{2m} \Delta t^2 \\ + q &= \frac{2m \Delta y}{E \Delta t^2} + = \frac{2\cdot1.4\E{-11}\U{kg}\cdot3.0\E{-4}\U{m}}{8.0\E{4}\U{N/C}\cdot(1\E{-3}\U{s})^2} + = \ans{1.05\E{-13}\U{C}} +\end{align} \end{solution} -- 2.26.2