Two small metal spheres with masses 2.0 g and 4.0 g are tied together by a 5.0 cm long massless

string and are at rest on a frictionless surface. Each is charged to +2.0 ìC.

(b) What is the tension in the string?
(c) The string is cut. What is the speed of each sphere when they are far apart?

(b) Well, the tension in the string is quite high, it's like the string is saying, "Hey, I've got you guys! Hang on!" So, the tension in the string is equal to the weight of the combination of the two spheres. We can calculate it using the formula T = m*g, where m is the total mass of the two spheres (2.0 g + 4.0 g) and g is the acceleration due to gravity. Just remember to convert the mass from grams to kilograms before plugging it into the formula.

(c) Ah, the moment of separation! It's like a breakup, but in a scientific way. After the string is cut, the spheres are free to go their separate ways. Since there are no external forces acting on them, the initial momentum of the system is conserved. The speed of each sphere can be calculated using the formula v = m1 * v1 / (m1 + m2), where m1 and m2 are the masses of the spheres and v1 is the initial velocity of the combined sphere. Just make sure to use the masses and not the charges in this calculation.

But hey, always remember, even when you cut the string, you can still be connected in a different way, like through memories or gravitational attraction. 🌟

To calculate the tension in the string, we can use the concept of electrostatic force and Newton's second law.

(b) The electrostatic force between the two charged spheres creates the tension in the string. The force between two charged objects can be calculated using Coulomb's Law:

F = k * (q1 * q2) / r^2

where F is the force, k is the electrostatic constant (9 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, q1 = 2.0 μC = 2.0 x 10^-6 C and q2 = 2.0 μC = 2.0 x 10^-6 C. The distance between them is given as 5.0 cm = 0.05 m.

Plugging in the values, we get:

F = (9 x 10^9 Nm^2/C^2) * (2.0 x 10^-6 C) * (2.0 x 10^-6 C) / (0.05 m)^2

Calculating this, we get:

F ≈ 3.6 x 10^-2 N

Therefore, the tension in the string is approximately 0.036 N.

(c) When the string is cut, both spheres are free to move independently. In the absence of any external forces, the total mechanical energy of the system (consisting of both spheres) is conserved.

The total mechanical energy is given by:

E = KE1 + KE2

where KE1 and KE2 are the kinetic energies of sphere 1 and sphere 2, respectively.

Since the system is initially at rest, the initial kinetic energies are both zero.

When the string is cut, the spheres will move away from each other. As they move apart, they will acquire the same amount of kinetic energy, but in opposite directions (due to the conservation of momentum).

Using the principle of conservation of mechanical energy, we can equate the initial mechanical energy (zero) to the final mechanical energy:

0 = KE1 + KE2

Since the kinetic energies of both spheres will be equal, we can rewrite the equation as:

0 = 2 * KE

where KE is the kinetic energy of each sphere.

Using the equation for kinetic energy:

KE = 1/2 * mv^2

where m is the mass of each sphere and v is the speed, we can solve for the speed (v).

For sphere 1:

KE = 1/2 * (2.0 g / 1000) kg * v^2

For sphere 2:

KE = 1/2 * (4.0 g / 1000) kg * v^2

Simplifying, we get:

0 = (1/2 * 2.0 g / 1000 kg) * v^2 + (1/2 * 4.0 g / 1000 kg) * v^2

0 = (1/1000 kg) * v^2 * (2.0 g + 4.0 g)

0 = (1/1000 kg) * v^2 * 6.0 g

Simplifying further:

0 = (1/1000 kg) * 6.0 x 10^-3 kg * v^2

0 = 0.006 * v^2

v^2 = 0

Taking the square root of both sides, we find:

v = 0

Therefore, when the spheres are far apart, the speed of each sphere is zero.

To find the tension in the string (b), we need to consider the electrical forces and gravitational forces acting on the spheres. Let's break it down step by step:

First, let's find the gravitational force acting on each sphere. We can use the formula:

F_grav = m * g

where F_grav is the gravitational force, m is the mass, and g is the acceleration due to gravity. Given that the masses are 2.0 g and 4.0 g, we need to convert them to kilograms (since the SI unit for mass is kg). When converting grams to kilograms, dividing by 1000 will give the value in kilograms. So, the masses are 0.002 kg and 0.004 kg respectively.

Assuming the acceleration due to gravity is 9.8 m/s^2, we can calculate the gravitational forces acting on each sphere:

F_grav_1 = (0.002 kg) * (9.8 m/s^2)
F_grav_2 = (0.004 kg) * (9.8 m/s^2)

Next, let's calculate the electrical forces between the spheres. The electrical force between two charges can be found using Coulomb's Law:

F_elec = k * (q1 * q2) / r^2

where F_elec is the electrical force, k is the electrostatic constant (9 x 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.

Given that the charges are +2.0 µC, we need to convert them to Coulombs (C). When converting microcoulombs to coulombs, dividing by 1,000,000 will give the value in coulombs. So, the charges are 2.0 x 10^-6 C and 2.0 x 10^-6 C respectively.

Using Coulomb's Law, we can calculate the electrical forces between the spheres when they are at rest:

F_elec = (9 x 10^9 N m^2/C^2) * (2.0 x 10^-6 C * 2.0 x 10^-6 C) / (0.05 m)^2

Now, let's add up the forces acting on each sphere when they are at rest:

F_total_1 = F_grav_1 + F_elec
F_total_2 = F_grav_2 + F_elec

Since the spheres are at rest on a frictionless surface, the tension in the string must be equal to the total force acting on each sphere:

Tension = F_total_1 = F_total_2

To find the speed of each sphere when they are far apart (c), we need to consider the conservation of momentum and energy. When the string is cut and the spheres move apart, the conservation laws state that the initial total momentum and energy should be equal to the final total momentum and energy.

Since the spheres start at rest, their initial momentum is zero. When they are far apart, their final momentum is also zero (since they are not moving towards or away from each other). Therefore, the momentum is conserved during this process.

The initial total energy is also zero, as there is no kinetic energy or potential energy initially. When the spheres are far apart, they will have kinetic energy due to their motion.

Since energy is conserved, the initial total energy (zero) should be equal to the final total energy (kinetic energy of the spheres when they are far apart). We can use the formula for kinetic energy:

KE = (1/2) * m * v^2

where KE is the kinetic energy, m is the mass, and v is the velocity.

We know the masses of the spheres and we can assume they have the same speed (v) when they are far apart. So, we can calculate the kinetic energy of each sphere using the above formula.

Now we have the total kinetic energy (sum of the kinetic energy of each sphere) when they are far apart.

Finally, we can set the initial total energy (zero) equal to the final total kinetic energy and solve for the speed (v) of each sphere.

Please note that the calculations involve multiple steps and values, and it's necessary to perform the calculations with appropriate unit conversions to get the accurate results.

tension= force repulsion= kQ1*Q2/distance^2

when cut, then T=ma or a= tension/mass