Two metal spheres with masses 4.0 g and 6.0 g are tied together with 6.0 cm long massless string and are at rest on a frictionless surface. Each sphere is charged to + 5.0 nC.
a. What is the energy in the system?
b. What is the tension in the string?
c. The string is cut. What is the speed of the 4.0 g sphere when they are far apart?
The energy is equal to (kq^2)/r
To find the answers to these questions, we need to apply some principles of physics, namely the concepts of energy, tension, and conservation of momentum.
a. To calculate the energy in the system, we need to consider the potential energy and the electrostatic potential energy of the charged spheres. The potential energy of an object in a gravitational field is given by the equation E = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. However, in this case, we are dealing with electrostatic potential energy, not gravitational potential energy.
The electrostatic potential energy between two charged spheres is given by the equation E = k*q1*q2/r, where k is the Coulomb's constant (8.99 x 10^9 N*m^2/C^2), q1 and q2 are the charges of the two spheres (+5.0 nC and +5.0 nC), and r is the distance between the centers of the spheres (6.0 cm = 0.06 m).
E = k*q1*q2/r
E = (8.99 x 10^9 N*m^2/C^2) * (5.0 x 10^-9 C) * (5.0 x 10^-9 C) / (0.06 m)
E ≈ 1.498 x 10^-6 J
Therefore, the energy in the system is approximately 1.498 x 10^-6 J.
b. To find the tension in the string, we can consider the equilibrium of forces acting on the system. Since the spheres are at rest on a frictionless surface, the net force on each sphere must be zero.
For the 4.0 g sphere:
T - F_e = 0
T = F_e
For the 6.0 g sphere:
T + F_e = 0
T = -F_e
Where T is the tension in the string and F_e is the electrostatic force between the spheres. The electrostatic force can be calculated using Coulomb's law:
F_e = k*q1*q2/r^2
F_e = (8.99 x 10^9 N*m^2/C^2) * (5.0 x 10^-9 C) * (5.0 x 10^-9 C) / (0.06 m)^2
F_e ≈ 1.122 N
Thus, the tension in the string is approximately 1.122 N.
c. When the string is cut, the two spheres will move apart from each other due to the conservation of momentum. Since no external forces act on the system, the total momentum before cutting the string will be equal to the total momentum after the string is cut.
The momentum p of an object is given by the equation p = mv, where m is the mass and v is the velocity.
Before cutting the string:
p_initial = m1*v1 + m2*v2
After cutting the string, the 4.0 g sphere moves in one direction, and the 6.0 g sphere moves in the opposite direction. Let's assume the 4.0 g sphere moves to the right with velocity v_final and the 6.0 g sphere moves to the left with velocity -v_final. The masses of the spheres remain the same.
After cutting the string:
p_final = m1*v_final + m2*(-v_final)
According to the law of conservation of momentum, p_initial = p_final.
m1*v1 + m2*v2 = m1*v_final + m2*(-v_final)
Plug in the given values:
(4.0 g)*(0 m/s) + (6.0 g)*(0 m/s) = (4.0 g)*v_final + (6.0 g)*(-v_final)
Simplifying the equation:
0 = -2.0 g * v_final
v_final = 0 m/s
Therefore, the speed of the 4.0 g sphere when they are far apart is 0 m/s.