Dimensions of G, Gravitational Potential, PE and Field Intensity
Physics · Gravitation · NEET
Four gravitation quantities have four different dimensional formulas: G = [M⁻¹L³T⁻²], Gravitational PE = [ML²T⁻²] (same as energy), Gravitational Potential = [M⁰L²T⁻²] (PE per unit mass, so no M), and Field Intensity = [M⁰LT⁻²] (same as acceleration g). Memory hook: Potential is PE with the mass removed, and Field Intensity is just g in disguise, so it copies g's dimensions exactly.
The four gravitation quantities side by side. Notice Potential = PE with mass divided out (M drops to M⁰), and Intensity has the same dimensions as g [LT⁻²].
Your doubts, answered
How do I derive the dimensions of G from the formula?
Start from F = GMm/r². Rearrange for G: G = F·r² / (M·m). Now put in dimensions: Force [MLT⁻²], r² is [L²], and M·m (two masses) is [M²]. So G = ([MLT⁻²]·[L²]) / [M²] = [M L³ T⁻²] / [M²] = [M⁻¹ L³ T⁻²]. The two masses in the denominator are why G carries M to the power minus one.
Why does gravitational potential have NO mass in its dimensions but PE does?
Gravitational Potential Energy U = −GMm/r depends on the mass m of the body, so its dimensions are the same as energy, [ML²T⁻²]. Gravitational Potential V = U/m = −GM/r is energy PER UNIT mass, so you divide out one M. That gives V = [ML²T⁻²]/[M] = [M⁰L²T⁻²] = [L²T⁻²]. Rule: potential is per-unit-mass, so it always has M⁰.
Why does gravitational field intensity have the same dimensions as g?
Field intensity E = F/m = force per unit mass. That is exactly what acceleration due to gravity g is. So E has the dimensions of acceleration: [LT⁻²], written [M⁰LT⁻²]. Its SI unit is N/kg, which is the same as m/s². If a question asks for the dimensions of gravitational intensity, just write g's dimensions.
Is gravitational potential [L²T⁻²] the same as the square of velocity?
Dimensionally yes. Velocity is [LT⁻¹], so velocity² is [L²T⁻²], which matches gravitational potential and also matches escape velocity squared (v² = 2GM/R). This is why √(GM/R) has the dimensions of speed. NEET sometimes hides this: any quantity with [L²T⁻²] could be potential, latent heat, or velocity squared, so read the units given.
How can I quickly find dimensions of a combination like E/G?
Write each part separately, then divide the powers. Energy E = [ML²T⁻²], Gravitational constant G = [M⁻¹L³T⁻²]. E/G = [M^(1−(−1)) L^(2−3) T^(−2−(−2))] = [M²L⁻¹T⁰]. Handle each of M, L, T separately and subtract the powers. This exact trick was tested in NEET 2021.
⚠️ The NEET trap ✗ Gravitational potential and gravitational field intensity have the same dimensions because both come from gravity. ✓ They are different. Potential V = [L²T⁻²] (energy per mass). Field intensity E = [LT⁻²] (force per mass = acceleration). They differ by one power of L. Potential is a scalar energy-type quantity; intensity is a vector acceleration-type quantity. 🧠 Potential = energy per mass (L²T⁻²). Intensity = force per mass = g (LT⁻²). One extra L separates them.
Real NEET questions
NEET 2022
Match List-I with List-II.
List-I: (a) Gravitational constant G, (b) Gravitational potential energy, (c) Gravitational potential, (d) Gravitational intensity.
List-II: (i) [L²T⁻²], (ii) [M⁻¹L³T⁻²], (iii) [LT⁻²], (iv) [ML²T⁻²].
A · (a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)
B · (a)-(ii), (b)-(iv), (c)-(i), (d)-(iii) ✓
C · (a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)
D · (a)-(iv), (b)-(ii), (c)-(i), (d)-(iii)
Solution: Take each quantity one at a time.
G from F = GMm/r²: G = Fr²/(Mm) = [MLT⁻²][L²]/[M²] = [M⁻¹L³T⁻²] → (ii).
Gravitational PE, U = −GMm/r, is an energy: [ML²T⁻²] → (iv).
Gravitational potential V = U/m (energy per unit mass): [ML²T⁻²]/[M] = [L²T⁻²] → (i).
Gravitational intensity E = F/m (force per unit mass = acceleration): [LT⁻²] → (iii).
So (a)-(ii), (b)-(iv), (c)-(i), (d)-(iii). Answer B.
NEET 2021
If E and G respectively denote energy and gravitational constant, then E/G has the dimensions of:
A · [M][L⁰][T⁰]
B · [M²][L⁻²][T⁻¹]
C · [M²][L⁻¹][T⁰] ✓
D · [M][L⁻¹][T⁻¹]
Solution: Write both dimensional formulas.
Energy E = [ML²T⁻²].
Gravitational constant G = [M⁻¹L³T⁻²].
Divide, subtracting powers of M, L, T separately:
M: 1 − (−1) = 2
L: 2 − 3 = −1
T: −2 − (−2) = 0
So E/G = [M²L⁻¹T⁰]. Answer C.
Solved Gravitation NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
What is the dimensional formula of the universal gravitational constant G?
G = [M⁻¹L³T⁻²]. Its SI unit is N·m²/kg² (or equivalently m³ kg⁻¹ s⁻²), and its value is 6.67 × 10⁻¹¹ in SI units.
What are the dimensions of gravitational potential energy?
Gravitational PE has the dimensions of energy: [ML²T⁻²]. Its SI unit is the joule (J). This is because U = −GMm/r is a form of energy.
What are the dimensions of gravitational potential?
Gravitational potential V = [M⁰L²T⁻²] = [L²T⁻²]. It is energy per unit mass, so the mass dimension cancels out. Its SI unit is J/kg.
What are the dimensions of gravitational field intensity?
Gravitational field intensity E = [M⁰LT⁻²] = [LT⁻²], the same as acceleration due to gravity g. Its SI unit is N/kg, which equals m/s².
Why are these dimensions important for NEET?
NEET regularly asks match-the-column and combination questions (like E/G or hc/G) that need exact dimensional formulas. Knowing G, PE, potential and intensity by heart lets you solve these in seconds without full derivations.
Do gravitational potential and velocity squared have the same dimensions?
Yes. Both are [L²T⁻²]. This is why escape velocity v = √(2GM/R) works dimensionally: 2GM/R has dimensions [L²T⁻²], and its square root gives speed [LT⁻¹].
