The gravitational force F between two point masses m_1 and m_2 separated by a distance r i… — JAMB UTME

The gravitational force $F$ between two point masses $m_1$ and $m_2$ separated by a distance $r$ is expressed by the equation $F = \frac{G m_1 m_2}{r^2}$ , where $G$ represents the universal gravitational constant. What is the dimensional formula for $G$ ?

$M^{-1} L^3 T^{-2}$ Answer
B
$M L^3 T^{-2}$
C
$M^{-1} L^2 T^{-2}$
D
$M^{-1} L^3 T^{-1}$

Answer

The dimensional formula for the universal gravitational constant

G

M^{-1} L^3 T^{-2}

Isolating

G

gives

G = \frac{F r^2}{m_1 m_2}

. Substituting the fundamental dimensions for force (

M L T^{-2}

), distance (

L

), and mass (

M

) yields

\frac{(M L T^{-2})(L^2)}{M^2} = M^{-1} L^3 T^{-2}

Step-by-Step Solution

Make

G

the subject of the formula in Newton's law of gravitation.

G = \frac{F r^2}{m_1 m_2}

To derive the dimensions of

G

, isolate it in terms of force, distance, and mass.

Substitute fundamental dimensions for force, distance, and mass.

[G] = \frac{[F][r]^2}{[m_1][m_2]} = \frac{(M L T^{-2})(L^2)}{M \cdot M}

Force has dimensions

M L T^{-2}

, distance has dimension

L

, and mass has dimension

M

Simplify the powers of fundamental dimensions

M

L

, and

T

[G] = M^{1-2} L^{1+2} T^{-2} = M^{-1} L^3 T^{-2}

Applying exponent rules simplifies the combined base dimensions.

Key Concept

Dimensions of Physical Constants

Estimated Time:1m 15s

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