1 \begin{problem*}{25.27}
2 Two insulating spheres have radii $0.300\U{cm}$ and $0.500\U{cm}$,
3 masses $0.100\U{kg}$ and $0.700\U{kg}$, and uniformly distributed
4 charges $-2.00\U{$\mu$C}$ and $3.00\U{$\mu$C}$. They are released
5 from rest when their centers are separated by $1.00\U{m}$. \Part{a}
6 How fast will each be moving when they collide? \Part{b} \emph{What
7 if?} If the spheres were conductors, would the speeds be greater
8 or less than those calculated in part \Part{a}? Explain.
15 p_i = 0 &= p_f = m_1 v_1 - m_2 v_2 \\
17 v_2 &= \frac{m_1}{m_2} v_1 \;.
20 The electric potential energy is initially
22 U_i = qV_i = k_e \frac{q_1q_2}{r_{12,i}} \;,
24 and the final electric potential energy is
26 U_f = qV_f = k_e \frac{q_1q_2}{R_1+R_2} \;.
30 E_i = U_i &= E_f = U_f + K_{f,1} + K_{f,2} \\
31 K_{f,1} + K_{f,2} &= U_i - U_f \\
32 \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2
33 &= k_e \frac{q_1q_2}{r_{12,i}} - k_e \frac{q_1q_2}{R_1+R_2} \\
34 \frac{1}{2}\p({m_1 v_1^2 + m_2 \p({\frac{m_1}{m_2}v_1})^2})
35 &= k_e q_1 q_2 \p({\frac{1}{r_{12,i}} - \frac{1}{R_1+R_2}}) \\
36 \p({m_1 + \frac{m_1^2}{m_2}}) v_1^2
37 &= 2 k_e q_1 q_2 \p({\frac{1}{r_{12,i}} - \frac{1}{R_1+R_2}}) \\
38 \frac{m_1m_2 + m_1^2}{m_2} v_1^2
39 &= 2 k_e q_1 q_2 \p({\frac{1}{r_{12,i}} - \frac{1}{R_1+R_2}}) \\
40 v_1^2 &= \frac{2 k_e q_1 q_2 m_2}{m_1m_2 + m_1^2}
41 \p({\frac{1}{r_{12,i}} - \frac{1}{R_1+R_2}}) \\
42 v_1 &= \sqrt{\frac{2 k_e q_1 q_2 m_2}{m_1m_2 + m_1^2}
43 \p({\frac{1}{r_{12,i}} - \frac{1}{R_1+R_2}})}
44 = \ans{10.8\U{m/s}} \\
45 v_2 &= \frac{m_1}{m_2}v_1
46 = \sqrt{\frac{2 k_e q_1 q_2 m_1}{m_2^2 + m_1m_2}
47 \p({\frac{1}{r_{12,i}} - \frac{1}{R_1+R_2}})}
52 The speeds for conductors would be \ans{greater}, because the opposite
53 charges on each sphere would be drawn to the inner walls, reducing the
54 effective distance between the attracting charges and increasing the
57 Thinking about the problem in terms of electric potential, the
58 difference between the initial and final effective distances would be
59 greater ($\Delta r_{12}' = |r_{12,f}' - r_{12,i}'| > \Delta r_{12}$)
60 because as the spheres close the charge polarization will intensify.
61 This alone would increase the electric potential converted to kinetic
62 energy, but the distance change also occurs over a more sensitive
63 portion of the $1/r$ voltage curve.
68 size(6cm, 4cm, IgnoreAspect);
82 draw((r12i, V(r12i))--(r12f, V(r12i))--(r12f, V(r12f)), blue);
83 draw((r12ip, V(r12ip))--(r12fp, V(r12ip))--(r12fp, V(r12fp)), green);
84 draw(graph(V, r_min, r_max), red);
85 label("$\Delta r_{12}$", ((r12i+r12f)/2, V(r12i)), align=S, p=blue);
86 label("$\Delta r_{12}'$", ((r12ip+r12fp)/2, V(r12ip)), align=SW, p=green);
87 label("$\Delta V_{12}$", (r12f, (V(r12i)+V(r12f))/2), align=W, p=blue);
88 label("$\Delta V_{12}'$", (r12fp, (V(r12ip)+V(r12fp))/2), align=W, p=green);
89 xaxis("$r$", xmin=0, xmax=r_max, LeftTicks(NoZero));
90 yaxis("$\frac{V}{kq}$", ymin=0, ymax=V(r_min), RightTicks);