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?
a. To determine the energy in the system, we need to consider the potential energy of the charged spheres due to their charges and the gravitational potential energy due to their masses.
The potential energy due to the charges of the spheres can be calculated using the formula:
E_charge = (k * q1 * q2) / r
Where:
k is the Coulomb's constant (9 * 10^9 Nm^2/C^2)
q1 and q2 are the charges of the spheres (+5.0 nC each)
r is the distance between the spheres (6.0 cm)
Converting the charges to coulombs:
q1 = 5.0 nC = 5.0 * 10^-9 C
q2 = 5.0 nC = 5.0 * 10^-9 C
Converting the distance to meters:
r = 6.0 cm = 6.0 * 10^-2 m
Calculating the charge potential energy using the formula:
E_charge = (9 * 10^9 Nm^2/C^2 * 5.0 * 10^-9 C * 5.0 * 10^-9 C) / 6.0 * 10^-2 m
b. To determine the tension in the string, we need to consider the forces acting on each sphere. The tension in the string is equal to the force required to keep the spheres moving in a circle.
Since the spheres are at rest on a frictionless surface, the only forces acting on them are the tension in the string and the gravitational force due to their masses.
The gravitational potential energy can be calculated using the formula:
E_gravity = m * g * h
Where:
m is the mass of each sphere (4.0 g and 6.0 g)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height of the spheres (which is zero since they are on a frictionless surface)
Calculating the gravitational potential energy for each sphere and summing them will give us the total energy in the system.
E_gravity = (4.0 g * 9.8 m/s^2 * 0) + (6.0 g * 9.8 m/s^2 * 0)
c. When the string is cut, the two spheres will move apart due to their mutual electrostatic repulsion. To find the speed of the 4.0 g sphere when they are far apart, we need to apply the law of conservation of energy.
Before the string is cut, the total mechanical energy of the system is equal to the sum of the potential energy and kinetic energy of the spheres. After the string is cut, the potential energy is zero. Therefore, the initial total energy is equal to the final kinetic energy of the spheres.
Assuming that the final kinetic energy is fully transferred to the 4.0 g sphere, we can solve for its speed using the formula for kinetic energy:
E_initial = E_final
E_gravity + E_charge = (1/2) * m * v^2
Where:
m is the mass of the 4.0 g sphere (4.0 g)
v is the final velocity of the 4.0 g sphere
Solving for v will give us the speed of the 4.0 g sphere when they are far apart.