Question

Two small metallic spheres, each of mass m = 0.50 g, are suspended as pendulums by light strings from a common point as shown in the figure below. The spheres are given the same electric charge, and it is found that they come to equilibrium when each string is at an angle of 3.9° with the vertical. If each string has length L = 28.0 cm, what is the magnitude of the charge on each sphere? in Need Help? Waich N

Answer 1

f: Force from electric repulsion on one sphere
w: Weight of sphere
m: Mass of sphere
g: acceleration due to gravity
T: Tension in a string
ℓ: Length of string


The sum of the forces on a sphere along the horizontal axis is zero:
Tsin( θ ) = f

The sum of the forces on a sphere along the vertical axis is zero:
Tcos( θ ) = w


Tsin( θ ) / Tcos( θ ) = f / w

sin( θ ) / cos( θ ) = f / w

tan( θ ) = f / w
tan( θ ) = [ kₑ Q₁Q₂ / r² ] / mg
tan( θ ) = [ kₑQ² / r² ) ] / mg
tan( θ ) = kₑQ² / mgr²

Q² = mgr² tan( θ ) / kₑ
Q² = mg( 2ℓsin( θ ) )² tan( θ ) / kₑ
Q² = 4mgℓ²sin²( θ )tan( θ ) / kₑ
Q² = 4( 0.0005 kg)( 9.8 m/s² )( 0.28m )²sin²( 3.9º ) tan( 3.9º ) / 8.98755 × 10^9 N(m/C)²

Q² = 5.3920 × 10^17 C²
Q = 734302390 C