# Suppose you had two small boxes, each containing 1.0 g of protons. (a) If one were placed on the moon by an astronaut and the other were left on the earth, and if they were connected by a very light (and very long!) string, what would be the tension in the string? Express your answer in newtons and pounds. Do you need to take into account the gravitational forces of the earth and moon on the protons? Why? (b) What gravitational force would each box of protons exert on the other box?

## Calculate the number of protons that have a mass of 1 g. (Get that by dividing 1 by the number of grams per proton). Then calculate the charge of that amount of protons by multiplying by the proton charge.

Charge = 1 g * (1.602*10^-19C)/(1.672*10^-24 g)= 9.58*10^4 C

Use Coulomb's law to get the repulsion force when they are separated by the earth-moon distance.

For the weight force exerted by the earth and moon, use Newton's law of gravity or just multiply the total proton mass by g (for the earth) or g/6 (for the moon). I susepect they will be nelgligible compared to the electrospatic force. The same goes for the gravity force of the boxes on each other.

Do these calculations yourself; it will be a useful exercise for you.

## To determine the tension in the string, we need to consider the gravitational forces acting on each box of protons. We can use Newton's Law of Universal Gravitation, which states that the force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

(a) The tension in the string can be found by balancing the gravitational forces acting on the two boxes. We can assume the string is light and does not contribute significantly to the tension.

Let's calculate the tension step by step:

Step 1: Determine the gravitational force on each box of protons due to the moon and the earth.

The gravitational force acting on an object can be calculated using the formula:

F = (G * m1 * m2) / r^2

Where:
F is the gravitational force,
G is the universal gravitational constant (approximately 6.67430 × 10^-11 N(m/kg)^2),
m1 and m2 are the masses of the objects being attracted (in this case, the boxes of protons),
and r is the distance between the objects (the radius of the moon or the distance between the earth and moon).

The mass of an individual proton is approximately 1.67 x 10^-27 kg.

Let's assume the distance between the earth and moon is approximately 384,400 km (2.39 × 10^8 miles).

For the box of protons on the moon:
m1 = 1.0 g = 1.0 x 10^-3 kg (converting grams to kilograms)
m2 = mass of the moon (let's ignore the mass of the box)
r = radius of the moon (let's assume it is 1,737 km, which is the average radius of the moon)

For the box of protons on earth:
m1 = 1.0 g = 1.0 x 10^-3 kg (converting grams to kilograms)
m2 = mass of the earth (let's ignore the mass of the box)
r = distance between the earth and moon (384,400 km or 2.39 × 10^8 miles)

Step 2: Calculate the gravitational force on each box using the formula mentioned above.

For the box of protons on the moon:
F_moon = (G * m1 * m_moon) / r_moon^2

For the box of protons on earth:
F_earth = (G * m1 * m_earth) / r_earth^2

Step 3: Subtract the gravitational force on the box on the moon from the gravitational force on the box on earth to determine the net force.

Net force = F_earth - F_moon

This net force will be balanced by the tension in the string.

Step 4: Calculate the weight of one box of protons on earth.

Weight = mass * acceleration due to gravity

The acceleration due to gravity on earth is approximately 9.81 m/s^2.

Weight_earth = m1 * 9.81

Now let's calculate all the values:

- Mass of the moon: approximately 7.35 × 10^22 kg
- Mass of the earth: approximately 5.97 × 10^24 kg
- Radius of the moon: approximately 1,737 km or 1.74 × 10^6 m
- Distance between the earth and moon: approximately 384,400 km or 3.84 × 10^8 m
- Mass of one proton: approximately 1.67 × 10^-27 kg
- Weight of one box of protons on earth: m1 * 9.81

Using these values, we can calculate the tension in the string (net force) and the weight of one box of protons on earth.

(b) The gravitational force between the two boxes of protons can be calculated using the same formula mentioned above:

F_protons = (G * m1 * m2) / r^2

Where m1 and m2 represent the masses of the two boxes of protons and r is the distance between them.

Since the mass of each box is 1.0 g (1.0 x 10^-3 kg), you can substitute these values into the formula to calculate the gravitational force between them.

Please note that the weight of one box of protons on the moon would be negligible due to the moon's weaker gravitational pull compared to the earth. However, the gravitational force between the two boxes of protons themselves should still be considered.

Now you have all the information and steps needed to calculate the tension in the string and the gravitational force between the boxes of protons.