Topic 2.6 Notes – Gravitational Force

1. Newton’s Law of Universal Gravitation

Gravity is a universal, attractive force between any two objects with mass.

The magnitude of that force is

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

• $G = 6.67 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$
• $m_1, m_2$ are the masses
• $r$ is the distance between their centers of mass

What this equation is saying

• Force is directly proportional to each mass
  • Double one mass → $F_g$ doubles
  • Double both → $F_g$ quadruples
• Force is inversely proportional to $r^2$
  • Double distance → force becomes $1/4$ as large
  • Triple distance → force becomes $1/9$

That inverse-square behavior is huge. Gravity drops off fast as distance increases.

Direction and where it acts

• Always attractive
• Always along the line connecting the centers of mass
• Can be treated as acting at each object’s center of mass
• Forces come in equal and opposite pairs (Newton’s 3rd law)

On FRQs, if they ask you to describe the gravitational interaction, mention both magnitude (inverse-square) and direction (along the line, attractive).

2. Gravitational Field and Weight

Instead of thinking about two objects pulling on each other, we often describe gravity using a field.

A field tells you what would happen to a small test object placed at some point in space.

Gravitational Field

By definition,

$$g = \frac{F_g}{m}$$

Units: N/kg, which is the same as m/s².

For a spherical mass $M$:

$$g = G \frac{M}{r^2}$$

Notice what’s missing. The test mass canceled out. That’s why all objects fall at the same rate in the same place.

If gravity is the only force acting,

$$a = g$$

That connection between N/kg and m/s² is something AP loves to test conceptually.

Weight

Weight is the gravitational force on an object.

$$W = F_g = mg$$

• Mass (kg) is intrinsic.
• Weight (N) depends on the local gravitational field.

Near Earth’s surface, $g \approx 9.8\,\text{N/kg}$ and is often treated as constant.

When can we treat gravity as constant?

When the change in distance from Earth’s center is tiny compared to Earth’s radius.

That’s true for:

• Projectiles
• Ramps
• Elevators
• Buildings

Not true for:

• Satellites
• Large altitude changes

If the force doesn’t change significantly over the motion, you’re allowed to treat it as constant.

3. Apparent Weight and Acceleration

Your apparent weight is not $mg$. It’s the normal force.

On a Flat Surface

At rest:

$$F_N = mg$$

Scale reads true weight.

On an Incline

On a ramp, gravity still points straight down, but the normal force is perpendicular to the surface. Only the component of gravity perpendicular to the incline is balanced by the normal force.

$$F_N = mg \cos\theta$$

Free-body diagram of a block on an incline

In the diagram, $mg\cos\theta$ is perpendicular to the surface and $mg\sin\theta$ is parallel to it.

Since $mg\cos\theta < mg$, you “weigh less” on a ramp.

In an Accelerating Elevator

Apply Newton’s 2nd law vertically.

• Accelerating upward:

  $$F_N = mg + ma$$

  You feel heavier.

• Accelerating downward:

  $$F_N = mg - ma$$

  You feel lighter.

If $a = g$ downward:

$$F_N = 0$$

That’s weightlessness.

Students often forget to define a direction and write $\Sigma F = ma$ before plugging in. Always start with the free-body diagram.

Weightlessness

Weightless does not mean no gravity.

It means:

• No normal force, or
• Gravity is the only force acting (free fall)

Astronauts in orbit are constantly falling toward Earth, but moving sideways fast enough to miss it. No normal force → no apparent weight.

Equivalence Principle

If you’re in a closed box:

• You cannot tell whether you feel heavy because
  • You’re in gravity, or
  • The box is accelerating upward.

Acceleration and gravity are locally indistinguishable.

That idea connects directly to why $a = g$ is independent of mass.

4. Inertial Mass vs Gravitational Mass

These are conceptually different but experimentally equal.

Inertial Mass

Comes from Newton’s 2nd law:

$$F = ma$$

Measures resistance to acceleration.

More inertial mass → harder to accelerate.

Gravitational Mass

Appears in:

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

Determines:

• How strongly an object creates gravity
• How strongly it responds to gravity

Why They Matter

Experiments show:

$$m_{\text{inertial}} = m_{\text{gravitational}}$$

Because they’re equal, all objects fall with the same acceleration in a uniform field.

If those masses were different, heavier objects would fall differently than lighter ones.

That equality is the deep reason $a = g$ does not depend on mass.