understand how the concept of gravitational potential leads to the gravitational potential energy of two point masses and use EP = –GMm / r

What is Gravitational Potential?

Think of gravitational potential as the “height” of a ball in a gravity field, but instead of height we talk about the energy per unit mass that a body would have at a certain distance from a massive object. It’s a handy way to keep track of how much work you’d need to move a mass in a gravitational field.

🔭 In equations we write it as φ (phi). For a point mass M the potential at a distance r is:

φ = -\frac{GM}{r}

Notice the negative sign: the closer you are to the mass, the more negative (and therefore “deeper”) the potential.

When you have a second mass m in that field, its gravitational potential energy (GPE) is simply the product of its mass and the potential it experiences:

U = m\,φ

Substituting the expression for φ gives the familiar formula:

\$U = -\frac{GMm}{r}\$

So the GPE depends on the masses involved and how far apart they are.

Problem: A 0.5 kg toy rocket is 10 m above the surface of a planet with mass 5 × 10²⁴ kg. What is its gravitational potential energy relative to the planet’s surface?

1. Identify the variables:
   

   • G = 6.67 × 10⁻¹¹ m³ kg⁻¹ s⁻²
   • M = 5 × 10²⁴ kg
   • m = 0.5 kg
   • r = 10 m

2. Plug into the formula:
   

   \$U = -\frac{GMm}{r}\$

3. Calculate:
   

   \$U = -\frac{(6.67 × 10^{-11})(5 × 10^{24})(0.5)}{10}\$

   \$U = -\frac{(6.67 × 5 × 0.5) × 10^{13}}{10}\$

   \$U = -\frac{16.675 × 10^{13}}{10} = -1.6675 × 10^{13}\,\text{J}\$

4. Interpretation: The rocket has a negative potential energy of about −1.7 × 10¹³ J relative to the planet’s surface.

💡 Tip: Remember the negative sign means the energy is “stored” in the attraction; the closer you get, the more negative it becomes.

• Always check the sign of the potential energy: it should be negative for attractive forces.
• When a problem asks for “energy relative to infinity”, set r → ∞ so U → 0.
• Use the formula U = -GMm/r directly; no need to first find φ unless the question explicitly asks for it.
• Check units: G in m³ kg⁻¹ s⁻², masses in kg, distance in m → result in joules.
• For multiple masses, sum the energies: Utotal = Σ(-G Mi m / r_i).
• Remember that potential energy is a scalar; it doesn’t have a direction.

📝 Practice Question: Two masses, 2 kg and 3 kg, are 5 m apart. Calculate the gravitational potential energy of the system.