# Gravitational fields

> IB Physics SL · IB Physics SL 2025 Syllabus
> Source: https://www.owlsprep.com/study/ib-physics-sl-u4-gravitational-fields/

This module introduces the core concept of gravitational fields, which describes gravitational interaction between masses. You will learn to calculate field strength and potential for common mass distributions, and connect these two key quantities for exam problem solving.

**Prerequisites:** [Newton's universal law of gravitation](https://www.owlsprep.com/study/ib-physics-sl-u2-newtons-laws-of-motion/); [Circular motion kinematics](https://www.owlsprep.com/study/ib-physics-sl-u3-circular-motion/)

## Learning objectives

- Define gravitational field and gravitational field strength
- Calculate gravitational field strength for point and spherical masses
- State and apply the formula for gravitational potential
- Relate gravitational field strength to the gradient of gravitational potential

## What is a Gravitational Field?

**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. $$g = \frac{29.4\ N}{3.0\ kg} = 9.8\ N kg^{-1}$$

*Calculator:* allowed

## Gravitational Field Strength for Point Masses

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.

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

*Notation:* g

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

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

> **info**
>
> Near Earth's surface, the change in $r$ is negligible compared to Earth's radius, so we approximate the field as **uniform** with $g \approx 9.8\ N\ kg^{-1}$.

**Worked example:** Calculate the gravitational field strength at Earth's surface. Use $G = 6.67 \times 10^{-11}\ N\ m^2\ kg^{-2}$, Earth mass $M = 5.97 \times 10^{24}\ kg$, Earth radius $R = 6.37 \times 10^6\ m$.

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. $$g = \frac{(6.67 \times 10^{-11})(5.97 \times 10^{24})}{(6.37 \times 10^6)^2} \approx 9.81\ N\ kg^{-1}$$

*Calculator:* allowed

## Gravitational Potential

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.

**Gravitational potential** — 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:

*Notation:* V_g

$$V_g = -\frac{GM}{r}$$

> **note**
>
> Gravitational potential is always negative for finite $r$, because gravity is attractive. You need to do positive work to move a mass from a finite $r$ to infinity (where potential is zero).

**Worked example:** Calculate the gravitational potential at the surface of the Moon. Use Moon mass $M = 7.34 \times 10^{22}\ kg$, Moon radius $R = 1.74 \times 10^6\ m$, $G = 6.67 \times 10^{-11}\ N\ m^2\ kg^{-2}$.

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

*Calculator:* allowed

## Relation Between Field Strength and Potential

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. $$g = -\frac{dV_g}{dr} = -\frac{GM}{r^2}$$

**Check your understanding**

Test your understanding:

1. What is the magnitude of gravitational field strength at a point where the potential gradient is $4.2 \times 10^{-5}\ J\ kg^{-1}\ m^{-1}$?

   - $0$
   - $2.1 \times 10^{-5}\ N\ kg^{-1}$
   - $4.2 \times 10^{-5}\ N\ kg^{-1}$
   - $8.4 \times 10^{-5}\ N\ kg^{-1}$

   *Answer:* $4.2 \times 10^{-5}\ N\ kg^{-1}$

   *Why:* Correct. The magnitude of $g$ equals the magnitude of the potential gradient.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting the negative sign for gravitational potential
  - Why it fails: 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: Always include the negative sign in $V_g = -\frac{GM}{r}$ unless only the magnitude is requested
- **Wrong:** Using $g = \frac{GM}{r^2}$ for points inside a spherical mass
  - Why it fails: 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: Only use the inverse square relation for points outside the mass of the sphere
- **Wrong:** Confusing gravitational potential and gravitational potential energy
  - Why it fails: 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: Remember the relation: $U = m V_g$, where $U$ is potential energy and $V_g$ is potential
- **Wrong:** Adding field strengths as scalars when multiple masses are present
  - Why it fails: Gravitational field strength is a vector, so you must add components vectorially, not just add magnitudes
  - Correct: Break each field into x and y components, add components, then find the resultant vector magnitude and direction

## Cheatsheet

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

## 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](https://www.owlsprep.com/study/ib-physics-sl-u4-electric-fields/)
- [Magnetic fields](https://www.owlsprep.com/study/ib-physics-sl-u4-magnetic-fields/)

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