Gravitational fields

IB Physics SL· Unit 4: Fields, Topic 1: Gravitational fields· 6 min read

1. What is a Gravitational Field?★★☆☆☆⏱ 10 min

✓ Calculator OK

📘 Definition

Gravitational field

A gravitational field is a region of space where any mass placed within the region experiences an attractive gravitational force. Field concepts describe non-contact forces without requiring direct contact between interacting masses.

Example:

Any object near Earth's surface is in Earth's gravitational field, and experiences a weight force

Gravitational field strength is defined as the force per unit mass acting on a small test mass placed at the point of interest. The formula for gravitational field strength is:

g = \frac{F}{m}

📐 Worked Example

A 3.0 kg test mass experiences a downward force of 29.4 N when placed at a point near Earth's surface. Calculate the gravitational field strength at this point.

1
Recall the definition of gravitational field strength:
2
$g = \frac{F}{m}$
3
Substitute the given values for force and mass:
4
$g = \frac{29.4\ N}{3.0\ kg} = 9.8\ N kg^{-1}$

2. Gravitational Field Strength for Point Masses★★★☆☆⏱ 15 min

✓ Calculator OK

From Newton's law of universal gravitation, we can derive the gravitational field strength produced by a point mass $M$ at a distance $r$ from the center of the mass.

📘 Definition

Gravitational field strength (point mass)

g

For a point mass $M$ , the magnitude of gravitational field strength at distance $r$ is given by the inverse square law:

Example:

Spherical masses act like point masses with all mass concentrated at the center, for points outside the sphere

g = \frac{GM}{r^2}

📐 Worked Example

Calculate the gravitational field strength at Earth's surface. Use $G = 6.67 \times 1 0^{- 11} N m^{2} k g^{- 2}$ , Earth mass $M = 5.97 \times 1 0^{24} k g$ , Earth radius $R = 6.37 \times 1 0^{6} m$ .

1
Earth is spherical, so we can use the inverse square law for points at the surface:
2
$g = \frac{GM}{R^2}$
3
Substitute the values:
4
$g = \frac{(6.67 \times 10^{-11})(5.97 \times 10^{24})}{(6.37 \times 10^6)^2} \approx 9.81\ N\ kg^{-1}$

3. Gravitational Potential★★★☆☆⏱ 12 min

✓ Calculator OK

Gravitational potential is a scalar quantity that describes the energy associated with a point in a gravitational field. It is defined relative to a reference point of zero potential at infinity, where gravitational effects are zero.

📘 Definition

Gravitational potential

V_g

Gravitational potential at a point is the gravitational potential energy per unit mass of a test mass placed at that point. For a point mass $M$ , gravitational potential at distance $r$ is:

V_g = -\frac{GM}{r}

📐 Worked Example

Calculate the gravitational potential at the surface of the Moon. Use Moon mass $M = 7.34 \times 1 0^{22} k g$ , Moon radius $R = 1.74 \times 1 0^{6} m$ , $G = 6.67 \times 1 0^{- 11} N m^{2} k g^{- 2}$ .

1
Use the formula for gravitational potential for a spherical mass:
2
$V_g = -\frac{GM}{R}$
3
Substitute values:
4
$V_g = -\frac{(6.67 \times 10^{-11})(7.34 \times 10^{22})}{1.74 \times 10^6} \approx -2.82 \times 10^6\ J\ kg^{-1}$

4. Relation Between Field Strength and Potential★★★★☆⏱ 13 min

✓ Calculator OK

Gravitational field strength is the negative gradient of gravitational potential. This means the field strength equals the negative rate of change of potential with distance:

g = -\frac{dV_g}{dr}

The negative sign indicates that gravitational field points in the direction of decreasing potential, which is always towards the attracting mass.

📐 Worked Example

Derive the formula for gravitational field strength from the formula for gravitational potential for a point mass.

1
Start with the gravitational potential formula:
2
$V_g = -\frac{GM}{r}$
3
Differentiate $V_{g}$ with respect to $r$ :
4
$\frac{dV_g}{dr} = \frac{GM}{r^2}$
5
Substitute into the potential gradient relation:
6
$g = -\frac{dV_g}{dr} = -\frac{GM}{r^2}$

✓ Quick check

Test your understanding:

What is the magnitude of gravitational field strength at a point where the potential gradient is $4.2 \times 1 0^{- 5} J k g^{- 1} m^{- 1}$ ?
- $0$
- $2.1 \times 1 0^{- 5} N k g^{- 1}$
- $4.2 \times 1 0^{- 5} N k g^{- 1}$
- $8.4 \times 1 0^{- 5} N k g^{- 1}$
Reveal answer
2 —
Correct. The magnitude of $g$ equals the magnitude of the potential gradient.

5. Common Pitfalls

Wrong move:

Forgetting the negative sign for gravitational potential

Why:

Potential is defined with zero at infinity, and is always negative for all points near an attracting mass. Omitting the sign will lead to wrong energy calculations

Correct move:

Always include the negative sign in $V_{g} = - \frac{GM}{r}$ unless only the magnitude is requested

Wrong move:

Using $g = \frac{GM}{r ^{2}}$ for points inside a spherical mass

Why:

The inverse square law only applies to points outside the spherical mass. Inside the mass, field strength increases linearly with $r$ , not decreases with $\frac{1}{r ^{2}}$

Correct move:

Only use the inverse square relation for points outside the mass of the sphere

Wrong move:

Confusing gravitational potential and gravitational potential energy

Why:

Both terms sound similar but are different quantities: potential is per unit mass, while potential energy is the total energy for a given mass

Correct move:

Remember the relation: $U = m V_{g}$ , where $U$ is potential energy and $V_{g}$ is potential

Wrong move:

Adding field strengths as scalars when multiple masses are present

Why:

Gravitational field strength is a vector, so you must add components vectorially, not just add magnitudes

Correct move:

Break each field into x and y components, add components, then find the resultant vector magnitude and direction

6. Quick Reference Cheatsheet

Quantity	Formula	Key Notes
Gravitational field strength	$g = \frac{F}{m} = \frac{GM}{r ^{2}}$	Vector, units $N k g^{- 1}$ , points towards mass
Gravitational potential	$V_{g} = - \frac{GM}{r}$	Scalar, units $J k g^{- 1}$ , always negative
Field-potential relation	$g = - \frac{d V _{g}}{d r}$	g = negative potential gradient
Near Earth uniform field	$g \approx 9.8 N k g^{- 1}$	Approximately constant magnitude and direction

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 1
Gravitational field strength calculation
2022 · 2
Potential and field strength relation

Going deeper

syllabusIB Physics SL 2025 Syllabus

What's Next

Gravitational fields are the first core field concept you learn in IB Physics, and the patterns you master here generalize directly to electric fields later in the course. This sub-topic underpins all problems involving orbital motion of planets, moons, and artificial satellites, including escape speed calculation and orbital energy. A solid understanding of potential and field strength is essential for many long answer exam questions, and helps you connect energy and force concepts in fields. Explore the following related topics next:

→Electric fields
→Magnetic fields