Given Information:
Mass of sphere A = m1 = 5 g
Mass of sphere B = m2 = 10 g
Charge = q1 = q2 = q = 5 mC = 5x10⁻³ C
Length = d = 1 m
Required Information:
Potential energy = ?
Acceleration = ?
Speed = ?
Answer:
U = 2.247x10⁵ j
a1 = 4.49x10⁷ m/s²
a2 = 2.247x10⁷ m/s²
v1 = -7740.8 m/s
v2 = 3870.4 m/s
Explanation:
(a) What is the electric potential energy of the system?
The electric potential energy can be found using
U = q²/4πe₀d
Where 1/4πe₀ is equal to 8.99x10⁹
U = ( 5x10⁻³)²*8.99x10⁹/1
U = 2.247x10⁵ j
(b) Suppose you cut the string.At that instant, what is the acceleration of each sphere?
Sphere A:
As we know from Newton's second law of motion
F1 = m1a1
We need the force, that can be found using Coulomb's law
F1 = q²/4πe₀d²
F1 = (5x10⁻³)²*8.99x10⁹/1²
F1 = 2.247x10⁵ N
a1 = F1/m1
a1 = 2.247x10⁵/0.005
a1 = 4.49x10⁷ m/s²
Sphere B:
F2 = q²/4πe₀d²
F2 = (5x10⁻³)²*8.99x10⁹/1²
F2 = 2.247x10⁵ N
a2 = F2/m2
a2 = 2.247x10⁵/0.010
a2 = 2.247x10⁷ m/s²
(c) A long time after you cut the string, what is the speed of each sphere?
The potential energy would be converted into kinetic energy
0.5m1v1² + 0.5m2v2² = 2.247x10⁵
According to the equation of conservation of linear momentum
m1v1 + m2v2 = 0
v1 = -m2/m1(v2)
v1² = m2²/m1²(v2²)
0.5m1(m2²/m1²(v2²))
so
0.5v2²((m2²/m1)+m2) = 0.5v2²((0.01²/0.005)+0.01) = 0.5v2²(0.03)
0.5v2²(0.03) = 2.247x10⁵
v2² = 2.247x10⁵/0.015
v2 = √1.49x10⁷
v2 = 3870.4 m/s
v1 = -m2/m1(v2)
v1 = -0.01/0.005(3870.4)
v1 = -7740.8 m/s
The negative sign means that both spheres move in opposite directions.