Newton’s law of universal gravitation is F Gmm2/r², where F is the magnitude of the gravitational force exerted by one object on another, m₁ and m₂ are the masses of the ob-jects, and r is the distance between them. Determine the dimensions and the SI units of the proportionality constant G.
The Correct Answer and Explanation is :
Newton’s law of universal gravitation is given by the equation:
[
F = G \frac{m_1 m_2}{r^2}
]
Where:
- ( F ) is the gravitational force between two objects (measured in Newtons, N),
- ( G ) is the gravitational constant (the quantity we want to determine),
- ( m_1 ) and ( m_2 ) are the masses of the two objects (measured in kilograms, kg),
- ( r ) is the distance between the centers of the two masses (measured in meters, m).
Step 1: Dimensions of Gravitational Force
The gravitational force ( F ) is measured in Newtons, and the dimensional formula of force (from Newton’s second law, ( F = ma )) is:
[
[F] = M L T^{-2}
]
Where:
- ( M ) represents mass,
- ( L ) represents length,
- ( T ) represents time.
Step 2: Dimensions of the Other Quantities
The dimensional formulas for ( m_1 ), ( m_2 ), and ( r ) are:
- ( [m_1] = [m_2] = M ),
- ( [r] = L ).
Step 3: Rearranging the Equation
Now, substitute the dimensional formulas into Newton’s law of universal gravitation:
[
[F] = [G] \frac{[m_1][m_2]}{[r]^2}
]
Substituting the known dimensional formulas:
[
M L T^{-2} = [G] \frac{M^2}{L^2}
]
Solving for the dimensions of ( G ):
[
[G] = \frac{M L^3 T^{-2}}{M^2} = M^{-1} L^3 T^{-2}
]
Thus, the dimensions of the gravitational constant ( G ) are ( [G] = M^{-1} L^3 T^{-2} ).
Step 4: SI Units of ( G )
To find the SI units of ( G ), we use the fact that:
- ( M ) is measured in kilograms (kg),
- ( L ) is measured in meters (m),
- ( T ) is measured in seconds (s).
Therefore, the SI units of ( G ) are:
[
\text{SI units of } G = \frac{\text{kg}^{-1} \text{m}^3}{\text{s}^2}
]
Final Answer:
- The dimensions of ( G ) are ( M^{-1} L^3 T^{-2} ).
- The SI units of ( G ) are ( \text{kg}^{-1} \text{m}^3 \text{s}^{-2} ).
I will now generate an image to visualize the equation and the dimensions.
Here is an educational diagram illustrating Newton’s Law of Universal Gravitation, including the dimensions and SI units of the gravitational constant ( G ). Let me know if you’d like any adjustments or further explanation!
