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?
To answer these questions, we can use the principles of electrostatics and conservation of energy. Let's go through each question step by step.
a. To find the energy in the system, we can calculate the potential energy stored in the charged spheres. The formula for the potential energy of a charged object is given by U = (k * q1 * q2) / r, where U is the potential energy, k is the electrostatic constant (9.0 x 10^9 Nm^2/C^2), q1 and q2 are the charges of the two spheres, and r is the distance between them.
In this case, both spheres have a charge of +5.0 nC. The distance between them is given as 6.0 cm (0.06 m). Plugging these values into the formula, we get:
U = (9.0 x 10^9 Nm^2/C^2) * (5.0 x 10^-9 C) * (5.0 x 10^-9 C) / 0.06 m
Calculating this expression will give you the energy in the system.
b. To find the tension in the string, we need to consider the forces acting on the spheres. In this situation, the only force acting on each sphere is the electrostatic force due to the interaction between the charges. Since the system is at rest, the forces must balance each other.
The electrostatic force between two charged objects is given by F = (k * q1 * q2) / r^2, where F is the force, k is the electrostatic constant, q1 and q2 are the charges, and r is the distance between the charges.
Since the spheres are at rest, the tension in the string must be equal to the magnitude of the electrostatic force. Hence, the tension in the string will be the same for both spheres.
Plugging in the values of the charges and the distance, we get:
F = (9.0 x 10^9 Nm^2/C^2) * (5.0 x 10^-9 C) * (5.0 x 10^-9 C) / (0.06 m)^2
Calculating this expression will give you the tension in the string.
c. When the string is cut, the 4.0 g sphere will move away from the 6.0 g sphere due to the repelling forces between their like charges. To find the speed of the 4.0 g sphere, we can use conservation of energy.
The initial potential energy of the system will convert into kinetic energy as the spheres separate. We can set up an energy equation:
Initial potential energy = Final kinetic energy
Using the formula for gravitational potential energy, U = m * g * h, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height, we can write:
(m1 * g * h1) + (m2 * g * h2) = (1/2) * (m1 * v^2) + (1/2) * (m2 * v^2)
Since both spheres are on a frictionless surface, the heights h1 and h2 are zero.
(m1 * g * 0) + (m2 * g * 0) = (1/2) * (m1 * v^2) + (1/2) * (m2 * v^2)
Simplifying the equation, we get:
0 = (1/2) * (m1 * v^2) + (1/2) * (m2 * v^2)
Rearranging the equation, we get:
v^2 = (0 - m2 * g * 0) / (m1 + m2) [since m1 = 4.0 g and m2 = 6.0 g]
Taking the square root of both sides, we can find the speed (velocity) of the 4.0 g sphere when they are far apart.
Note: Make sure to convert the masses to kilograms (1 g = 0.001 kg) before using the equations.