latex/problems/Serway_and_Jewett_8/problem23.50.tex

   1 \begin{problem*}{23.50}
   2 A small sphere of charge $q_1=0.800\U{$\mu$C}$ hangs from the end of a
   3 spring as in Figure~P23.50a.  When another small sphere of charge
   4 $q_2=-0.600\U{$\mu$C}$ is held beneath the first sphere as in Figure
   5 P23.50b, the spring stretches by $d=3.50\U{cm}$ from its original
   6 length and reaches a new equilibrium position with a separation
   7 between the charges of $r=5.00\U{cm}$.  What is the force constant of
   8 the spring?
   9 \begin{center}
  10 \begin{asy}
  11 import Mechanics;
  12 import ElectroMag;
  13
  14 real u = 0.1cm;
  15 real ceil_width = 3u;
  16 real frame_space = 20u;
  17 real initial_stretch = 10u;
  18 real dx = 3.5u;
  19 real dab = 5u;
  20
  21 Surface c1 = Surface(pFrom=(ceil_width,0), pTo=(-ceil_width,0));
  22 Charge a1 = pCharge((0, -initial_stretch), L="$q_1$");
  23 Spring s1 = Spring(pFrom=(0,0), pTo=a1.center());
  24 c1.draw(); s1.draw(); a1.draw();
  25
  26 Surface c2 = Surface(pFrom=(frame_space+ceil_width,0),
  27                     pTo=(frame_space-ceil_width,0));
  28 Charge a2 = pCharge((frame_space, -initial_stretch-dx), L="$q_1$");
  29 Charge b = nCharge(a2.center() - (0, dab), L="$q_2$");
  30 Spring s2 = Spring(pFrom=(frame_space,0), pTo=a2.center());
  31 c2.draw(); s2.draw(); a2.draw(); b.draw();
  32 \end{asy}
  33 \end{center}
  34 \end{problem*}
  35
  36 \begin{solution}
  37 The addition downward force from electric attraction is balanced by
  38 the additional upward force from spring extension, so
  39 \begin{align}
  40   \kappa d = |F_\text{spring}| &= |F_e| = k\frac{|q_1 q_2|}{r^2} \\
  41   \kappa &= k\frac{|q_1 q_2|}{dr^2}
  42     = \ans{49.3\U{N/m}} \;.
  43 \end{align}
  44 \end{solution}