Question

Two small charged spheres hang from cords of equal length as shown in the figure and make small angles ?1 and ?2 with the vertical.

(a) If Q1 = Q, Q2 = 8Q, and m1 = m2 = m, determine the ratio ?1 / ?2.




(b) If Q1 = Q, Q2 = 8Q, and m1 = m, and m2 = 7m, determine the ratio ?1 / ?2.




(c) Estimate the distance between the spheres for each case. (Use l, Q, m, k, and g as appropriate.)

Answer 1

1. Charge ratios don't matter, F is the same for each particle. ?1/?2 = 1
2. Small angle approx. means ? can substitute for sin? or tan?.
Fhor = m1gsin?1 = m2gsin?2 ==> m1?1 = m2?2 ==>
?1/?2 = m2/m1 = 2
3. My stab at it; we don't know F, so r must be in terms of m, q and L.
define ? = ?1+?2
define m(eff) = m1m2/(m1+m2)
In case 1, m(eff) = m/2, in case 2, m(eff) = 2m/3.
The physical meaning of m(eff) is that a force F between m1 and m2 accelerates the distance between them with a = F/m(eff).
Fhor = kq1q2/r^2 = m(eff)gtan? = m(eff)g? = m(eff)gr/L
kq1q2/r^2 = m(eff)gr/L
Lkq1q2 = m(eff)gr^3
r = [Lkq1q2/(m(eff)g)]^(1/3)

Answer 2

part b= 4

Answer 3

let F be the force between two particles in both cases
and R the distance between them

F = k Q1Q2/r²

forces on m1 ?
T1sin?1 = F -----(1)

forces on m1 ?
T1cos?1 = m1g -------(2)

from (1) ÷ (2)
tan?1 = F/m1g -----(3)
using this equation we can get that
tan?2 = F/m2g -----(4)

from (3) ÷ (4)
tan?1/tan?2 = m2/m1 -----(5)

we can use equation (5) to answer both (a) and (b)

-------------
** tan?1/tan?2 = m2/m1

(a) when m2 = m1 = m
tan?1/tan?2 = m/m
tan?1/tan?2 = 1
tan?1 = tan?2
?1 = ?2
?1/?2 = 1


(b) when m1=m, m2=2m
tan?1/tan?2 = m/2m
tan?1/tan?2 = 1/2
let tan?1/1 = tan?2/2 = k
?1 = tan?¹(k)
?2 = tan?¹(2k)
?1/?2 = tan?¹(k)/tan?¹(2k)

since tan?1 = F/m1g = k
?1/?2 = [tan?¹(F/m1g)]/[tan?¹(2F/m1g)]
===== ======== ========= ========= ========== ==========
NOTE: coming this far down to get this weird answer seems unexpected here. so just leaving the answer as tan?1/tan?2 = 1/2 would do
===== ======== ========= ========= ========== ==========

(c)
tan?1 = F/m1g
using trigonometry
sin?1 = F/v[(m1g)²+F²]

horizontal distance between them
= L(sin?1+sin?2)
= L{ F/v[(m1g)²+F²] + F/v[(mg)²+F²] }
substitute values for F and m to get answers for each case