Gravitational Force
Introduced| English |
|---|
| gravity/ˈɡrævɪti/ |
| weight/weɪt/ |
| gravitational field strength/ˌɡrævɪˈteɪʃənl fiːld streŋθ/ |
The apple and the Moon feel the same pull
- Legend says a falling apple made Newton wonder: does the same pull reach the Moon?
- It does. One law of gravity 引力 governs both the apple and the orbit.
- Near Earth we feel it as weight 重量 — the downward force $mg$.
- Farther out, the same law becomes the universal $F = \dfrac{GMm}{r^2}$.
Weight near a surface
- Weight is the gravitational force on an object: $W = mg$, straight down.
- Here $g$ is the gravitational field strength 重力场强度, about $9.8\ \tfrac{\text{N}}{\text{kg}}$ at Earth's surface.
- So a $1\ \text{kg}$ mass weighs about $9.8\ \text{N}$.
- Weight is a force (newtons); mass is the amount of matter (kilograms).
Gravity on different worlds
Switch the planet and watch how a weaker or stronger g bends the same launch differently.
What is the weight of a $5\ \text{kg}$ bag on Earth, where $g = 9.8\ \tfrac{\text{N}}{\text{kg}}$? Answer in $\text{N}$.
$W = mg = 5 \times 9.8 = 49\ \text{N}$.
On a distant planet a $10\ \text{kg}$ rock weighs $25\ \text{N}$. What is $g$ there, in $\tfrac{\text{N}}{\text{kg}}$?
$g = W/m = 25/10 = 2.5\ \tfrac{\text{N}}{\text{kg}}$.
Universal gravitation
- Every mass attracts every other mass: $F = \dfrac{G M m}{r^2}$.
- $M$ and $m$ are the two masses, $r$ the distance between their centres, $G$ a constant.
- The bigger the masses, the stronger the pull; the farther apart, the weaker.
- This one formula holds apples, moons, planets and galaxies together.
Select all changes that would make the gravitational force between two objects stronger.
$F = GMm/r^2$ grows with either mass and shrinks with distance. Moving them apart weakens it.
The inverse-square law
- Gravity falls off as $\dfrac{1}{r^2}$ — the inverse-square law.
- Double the distance and the force drops to a quarter; triple it and it drops to a ninth.
- That is also why $g$ is weaker on a mountain top than at sea level.
- Field strength links to the law by $g = \dfrac{GM}{r^2}$.
If the distance between two masses doubles, the gravitational force between them becomes:
Gravity is inverse-square: $F \propto 1/r^2$, so doubling $r$ gives $1/2^2 = 1/4$ the force.
Gravitational force is proportional to $1/r^{\_\_}$. Fill in the power.
It is an inverse-square law: $F \propto 1/r^2$.
Weight is not mass. Mass ($\text{kg}$) never changes; weight ($\text{N}$) depends on the local $g$. On the Moon ($g \approx 1.6\ \tfrac{\text{N}}{\text{kg}}$) your mass is unchanged but you weigh about one-sixth as much.
An object has the same weight everywhere in the universe.
Weight $W = mg$ depends on the local $g$. Mass stays fixed, but weight changes from planet to planet.
Find the weight of a $5\ \text{kg}$ bag on Earth ($g = 9.8\ \tfrac{\text{N}}{\text{kg}}$).
- $W = mg = 5 \times 9.8 = 49\ \text{N}$, pointing straight down.
Take the same bag to the Moon and its weight drops to about $8\ \text{N}$ — but its mass is still $5\ \text{kg}$.
Gravity gives weight $W = mg$ near a surface and follows the universal law $F = GMm/r^2$ in general. It is an inverse-square force (double $r$ → quarter $F$), and $g = GM/r^2$. Mass (kg) is fixed; weight (N) changes with $g$.