recall and use Newton’s law of gravitation F = Gm1m2 / r2 for the force between two point masses

Gravitational Force Between Point Masses

Think of two planets or even two tiny balls. They pull on each other with a force that depends on how heavy they are and how far apart they are. This pull is called the gravitational force and it’s described by Newton’s famous law. 🌍🪐

Newton’s Law of Gravitation

The force between two point masses is proportional to the product of their masses and inversely proportional to the square of the distance between them. In symbols:

$$F = \frac{G\,m_1\,m_2}{r^2}$$

Where:

• $F$ = gravitational force (Newtons, N)
• $G$ = universal gravitational constant $≈ 6.674\times10^{-11}\,\text{N·m}^2\text{/kg}^2$
• $m_1, m_2$ = masses of the two objects (kg)
• $r$ = distance between their centres (m)

Key Symbols & Units

Analogy: The Invisible Tug‑of‑War ⚖️

Imagine two kids holding a rope. The heavier a child is, the stronger they pull. The farther apart they stand, the weaker the pull feels. That’s exactly what happens with gravity: the “rope” is invisible, but the pull (force) depends on how heavy each mass is and how far apart they are. The closer the kids, the tighter the rope feels – just like the $1/r^2$ part of the formula. 📏

Worked Example

Two astronauts, each with a mass of $70\,\text{kg}$, are standing on opposite sides of a small asteroid that is $10\,\text{m}$ apart. What is the gravitational force between them?

1. Identify the variables: $m_1 = 70\,\text{kg}$, $m_2 = 70\,\text{kg}$, $r = 10\,\text{m}$.
2. Insert into the formula: $$F = \frac{(6.674\times10^{-11})\,(70)(70)}{10^2}$$
3. Calculate: $$F = \frac{(6.674\times10^{-11})\,(4900)}{100} \approx 3.27\times10^{-9}\,\text{N}$$
4. Interpret: The force is extremely tiny – about the weight of a grain of sand on Earth. That’s why astronauts feel weightless in space. 🧑‍🚀

Exam Tips 📚

Remember the formula: $F = Gm_1m_2/r^2$ – keep the order of operations in mind (multiply first, then divide).

Units are key: Always check that masses are in kg, distance in m, and the result will be in N. If you mix units, the answer will be wrong even if the maths is correct.

Use the $1/r^2$ rule: If the distance doubles, the force becomes one‑quarter of what it was. This can help you estimate answers quickly.

Show your work: Write the formula, plug in the numbers, and show the calculation steps. Marking schemes look for clear reasoning, not just a final number.

Practice with different scenarios: Try calculating the force between Earth and the Moon, or between two cars on a road. The more you practice, the more comfortable you’ll be with the numbers and the formula. 🚀