Being an acceleration, g has the acceleration due to gravity dimensional formula [M⁰ L¹ T⁻²], usually shortened to [LT⁻²]. It is length per time squared, with no mass in it.
Near Earth's surface g is about 9.8 m/s² (9.81 m/s² is also used). It changes with height, depth and latitude, yet its bracket never changes, because every value of g is still an acceleration. This page derives [LT⁻²] two ways, shows why N/kg and m/s² are one unit, checks the pendulum formula and lists the slips. For the variation of g with height and depth, see the acceleration due to gravity formula.
Quick facts
Dimensional formula
[M⁰ L¹ T⁻²], short form [LT⁻²]Defining formulas
,Standard value
9.8 m/s² (9.81 m/s² also used)SI unit
m/s², equal to N/kgVector or scalar
Vector, pointing downChapter
Gravitation, Class 9 and 11
Two ways to get the acceleration due to gravity dimensional formula
From weight
Weight is W = mg, so g = W/m. Weight is a force, [MLT⁻²], and dividing by mass [M] removes the M, leaving [LT⁻²].
From the law of gravitation
At a planet's surface, g = GM/R². With G as [M⁻¹L³T⁻²], the planet's mass [M] cancels the M⁻¹, and dividing by R², [L²], takes L³ down to L¹. The result is again [LT⁻²].
Both routes agree, as they must. The second one shows why g depends on the planet: M and R change from world to world while the bracket stays put. The gravitational constant page has the G step.
m/s², N/kg and other units
Turn [LT⁻²] into base units and you get m s⁻², written m/s². Gravity questions also quote g as a field strength in N/kg. These are the same unit:
- 1 N/kg = (1 kg m s⁻²)/(1 kg) = 1 m s⁻².
- So g = 9.8 m/s² and g = 9.8 N/kg say the same thing.
Write g = 9.8 m/s² in CGS units and in feet per second squared
CGS: L¹ means one factor of 100, so 9.8 m/s² = 980 cm/s².
Feet: 1 ft = 0.3048 m, so 9.8 ÷ 0.3048 = 32.2 ft/s². Time is in seconds in both, so only the length factor changes.
Variations of g that keep its bracket
| Quantity | Symbol or formula | Note |
|---|---|---|
| Any linear acceleration | a = (v − u)/t | g is a special case |
| Gravitational field strength | E_g = F/m | Same quantity seen as a field |
| Value of g at height h | g(1 − 2h/R) for small h | The bracket in brackets is a pure number |
| Effective g in a lift | g ± a | A sum of two accelerations |
All the variations of g in Class 11 gravitation multiply g by a pure number or add another acceleration, so none of them changes the dimensions. The dimensional formula of acceleration page covers the general case.
The pendulum as a dimension test
Check: is T = 2π√(l/g) for a simple pendulum dimensionally correct?
Inside the root: length over g is [L] ÷ [LT⁻²] = [T²].
The square root gives [T], a time. 2π has no dimensions. The formula is consistent.
Numbers: a pendulum 1.0 m long has a period near 2.0 s (the root of 1.0/9.8 is 0.319, and 2π times that is 2.01), very close to a seconds pendulum, one whose swing from one side to the other takes one second.
The same test exposes a wrong version such as T = 2π√(g/l): inside the root you would get [T⁻²], and the root would be a frequency, not a period.
Slips on gravity questions
- Confusing g with G. g is [LT⁻²]; G is [M⁻¹L³T⁻²]. Different letters, different quantities.
- Including mass because weight depends on mass. g is weight per unit mass; the M cancels.
- Thinking g at the poles has a different bracket. The value changes slightly, the dimensions do not.
- Writing N/kg and then calling it a force. Newton per kilogram is an acceleration.
The gravitation set is listed beside the other mechanics quantities on the dimensional formulas page.
Frequently asked questions
Is acceleration due to gravity a constant?
Not exactly. It is about 9.8 m/s² near Earth's surface but falls with height and depth and varies a little with latitude. Its dimensions, [LT⁻²], are always the same.
Does the dimensional formula for acceleration due to gravity include mass?
Only with a zero power: [M⁰ L¹ T⁻²]. Most books drop the M⁰ and print [LT⁻²].
Are m/s² and N/kg the same unit?
Yes. A newton is kg m/s², so dividing by a kilogram leaves m/s².
Is g a vector?
Yes. Near Earth it points toward the center of the Earth (centre of the Earth), straight down.
How do the dimensions of g and G differ?
g is an acceleration, [LT⁻²]. G is a universal constant, [M⁻¹L³T⁻²]. Multiplying G by a mass and dividing by a length squared turns one into the other.