Rearranging Newton's law of gravitation shows that the dimensional formula of gravitational constant is [M⁻¹ L³ T⁻²]. Mass appears with a negative power, length is cubed, and time sits squared in the denominator.
G is the universal constant in F = Gm₁m₂/r², with the value 6.674 × 10⁻¹¹ N m²/kg². It is the same everywhere, unlike g, which changes from place to place. Below: the derivation from the force law, the unit and its CGS value, how G differs from g, two orbit checks and the slips that cost marks. Force calculations between masses are on the gravitational force formula page.
Quick facts
Dimensional formula
[M⁻¹ L³ T⁻²]Defining law
Value
6.674 × 10⁻¹¹ N m²/kg²SI unit
N m²/kg² = m³ kg⁻¹ s⁻²Scalar or vector
Scalar constantChapter
Gravitation, Class 11
Deriving the dimensional formula of gravitational constant
- Newton's law of gravitation, with G made the subject, reads G = Fr²/(m₁m₂).
- Top: Fr² is [MLT⁻²] multiplied by [L²], or [ML³T⁻²].
- Bottom: two masses multiply to [M²].
- Divide: [ML³T⁻²] ÷ [M²]. Mass drops to M¹⁻² = M⁻¹, so G is [M⁻¹ L³ T⁻²].
The negative power of mass is the feature examiners like to test. It comes from the two masses in the denominator, which outweigh the single M in force.
The unit of G in SI and CGS
In base units, [M⁻¹ L³ T⁻²] reads kg⁻¹ m³ s⁻², usually written m³ kg⁻¹ s⁻². Grouped through the force law, the same unit is N m²/kg². Both are correct.
Express G = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² in CGS units
Use the powers from the bracket. L³: 1 m³ = 10⁶ cm³. M⁻¹: 1 kg⁻¹ = 10⁻³ g⁻¹. T⁻²: seconds are unchanged.
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So G in CGS is 6.674 × 10⁻⁸ dyne cm²/g². The negative power on M is why the factor 10³ divides rather than multiplies.
G next to g and other gravitation quantities
No common school quantity shares G's bracket, so a table of look alikes is more useful here. The trap is the letter g.
| Quantity | Symbol | Dimensions | Constant? |
|---|---|---|---|
| Universal gravitational constant | G | [M⁻¹L³T⁻²] | Yes, everywhere |
| Acceleration due to gravity | g | [LT⁻²] | No, varies with place |
| Gravitational potential | V = −GM/r | [L²T⁻²] | No |
| Product GM for a planet | GM | [L³T⁻²] | Fixed for that planet |
Multiplying G by a mass cancels the M⁻¹, which is why GM has pure length and time dimensions. The acceleration due to gravity page continues from there.
Checks from planets and orbits
Check: g = GM/R² and its value at Earth's surface
Dimensions: [M⁻¹L³T⁻²] × [M] ÷ [L²] = [LT⁻²], an acceleration. Consistent.
Value: GM for Earth is 3.98 × 10¹⁴ m³/s². Divide by R² = 4.06 × 10¹³ m² and g comes to 9.82 m/s², close to the usual 9.8 m/s².
Check: the orbit period relation T² = 4π²r³/(GM)
Right side: [L³] ÷ ([M⁻¹L³T⁻²] × [M]) = [L³] ÷ [L³T⁻²] = [T²].
The left side is a time squared, so the relation, Kepler's third law, passes.
Slips with big G
- Writing [ML³T⁻²]. The mass power must be −1; two masses sit below the line.
- Mixing up G and g. G is a universal constant with [M⁻¹L³T⁻²]; g is a local acceleration with [LT⁻²].
- Writing [M⁻²L²T⁻²]. That forgets the M in force. Do the force first, then divide.
- Converting to CGS by 10⁻³ alone. The length factor 10⁶ must come in too, giving 10³ overall.
The gravitation brackets sit with the rest of mechanics on the dimensional formulas page, and orbits are covered by orbital velocity.
Frequently asked questions
Why is the mass power negative in G?
Making G the subject puts m₁m₂, which is [M²], in the denominator. Force adds only one M on top, so the net power is −1.
Which SI unit does G have?
N m²/kg², which is the same as m³ kg⁻¹ s⁻² in base units.
Is G the same as g?
No. G is a universal constant, 6.674 × 10⁻¹¹ N m²/kg². g is the acceleration of free fall near a planet, about 9.8 m/s² on Earth. They have different dimensions.
Is G a scalar?
Yes. G is a scalar constant. The force it gives is a vector, pointing along the line joining the masses.
What is the value of G in CGS?
6.674 × 10⁻⁸ dyne cm²/g², obtained by putting the bracket powers on the unit conversion factors.