Newton's Law of Universal Gravitation
Every pair of masses in the universe attracts each other. Isaac Newton's law of universal gravitation quantifies that attraction: F = G × m1 × m2 / r², where F is the gravitational force in newtons, G is the gravitational constant (6.6743 × 10⁻¹¹ N·m²/kg²), m1 and m2 are the two masses in kilograms, and r is the distance between their centers of mass in meters. The force is always attractive, acts along the straight line connecting the two masses, and — by Newton's third law — pulls on both masses equally and oppositely, even though the resulting acceleration (a = F/m) differs for each one.
Why the distance is squared
Gravity is an inverse-square force: the r² in the denominator means force falls off much faster than distance grows. Doubling the separation between two masses reduces the force to one-quarter of its original value; moving them ten times farther apart cuts the force to one-hundredth. This inverse-square relationship also governs light intensity and other phenomena that spread outward from a point source across the surface of an expanding sphere (whose area grows as r²). It is why gravitational force between planets and moons — despite their enormous mass — is manageable at astronomical distances, while the same formula predicts negligible attraction between two coffee mugs sitting a few centimeters apart.
Getting accurate results
- Use center-to-center distance, not surface-to-surface. For large bodies like planets, r is measured between their centers of mass, not the gap between their surfaces.
- Keep units consistent. This calculator converts your chosen mass and distance units to kilograms and meters internally, so the result is always in newtons — mixing units yourself (e.g., mass in pounds with distance in feet) without conversion will give a wrong answer.
- Don't confuse gravitational force with weight. Weight (W = mg) is a shortcut for the gravitational force between an object and a specific planet at its surface; this calculator computes the general case between any two masses at any distance.
- Expect very large or very small numbers. Because G is tiny and astronomical masses are huge, results are often shown in scientific notation — that is expected, not an error.
Frequently Asked Questions
What is the formula for gravitational force?
Newton's law of universal gravitation states F = G × m1 × m2 / r², where G is the gravitational constant (6.6743 × 10⁻¹¹ N·m²/kg²), m1 and m2 are the two masses, and r is the distance between their centers. The force is attractive and acts along the line joining the two masses.
Why does distance have such a big effect on gravitational force?
Gravity follows an inverse-square law: force is divided by r², not r. Doubling the distance between two masses cuts the force to one-quarter of its original value; tripling the distance cuts it to one-ninth. This is why gravitational force drops off quickly as objects move apart.
Why don't I feel the gravitational pull of nearby objects?
You do, but it is extremely small. The gravitational constant G is tiny (6.6743 × 10⁻¹¹), so unless at least one mass is planet-sized, the resulting force is far too small to notice. Two 70 kg people standing 1 meter apart attract each other with only about 3.3 × 10⁻⁷ N — roughly the weight of a single grain of sand.
How is gravitational force different from weight?
Weight is the specific case of gravitational force between an object and a planet, usually written as W = mg, where g ≈ 9.8 m/s² at Earth's surface. That g value is actually G × M_earth / r_earth² simplified for objects near the surface — this calculator computes the more general F = Gm1m2/r² for any two masses at any distance.
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |