The SI unit of force, the kilogram-meter per second squared (kg · m/s2), is called the newton (N). Find the dimensions and the SI units of the constant G in Newton’s law of gravitation F = Gm1m2/r2. (For the dimension use L for length, M for mass, and T for time. For the units use m for meters, kg for kilograms, and s for seconds.)
The Correct Answer and Explanation is :
To find the dimensions and SI units of the gravitational constant ( G ) in Newton’s law of gravitation, we start with the equation:
[
F = G \frac{m_1 m_2}{r^2}
]
where:
- ( F ) is the gravitational force (in newtons, ( N )),
- ( m_1 ) and ( m_2 ) are the masses of the two objects (in kilograms, ( kg )),
- ( r ) is the distance between the centers of the two masses (in meters, ( m )),
- ( G ) is the gravitational constant.
Step 1: Identify the units and dimensions of each component.
- Force (( F )): The SI unit of force is the newton (N), which can be expressed in terms of base SI units:
[
1 \, N = 1 \, kg \cdot m/s^2
]
Thus, the dimensions of force are:
[
[F] = [M][L][T^{-2}]
] - Mass (( m_1 ) and ( m_2 )): The unit of mass is kilograms (( kg )):
[
[m] = [M]
] - Distance (( r )): The unit of distance is meters (( m )):
[
[r] = [L]
]
Step 2: Substitute into the equation to solve for ( G ).
Rearranging the equation for ( G ):
[
G = \frac{F r^2}{m_1 m_2}
]
Step 3: Substitute the dimensions and units into this expression.
Using the units and dimensions we identified:
[
[G] = \frac{[M][L][T^{-2}] \cdot [L^2]}{[M]^2}
]
This simplifies to:
[
[G] = \frac{[M][L^3][T^{-2}]}{[M^2]} = [M^{-1}][L^3][T^{-2}]
]
Step 4: Find the SI units of ( G ).
Substituting the base units:
[
[G] = \frac{m^3}{kg \cdot s^2}
]
Conclusion
The dimensions of the gravitational constant ( G ) are:
[
[G] = [M^{-1}][L^3][T^{-2}]
]
The SI units of ( G ) are:
[
[G] = \frac{m^3}{kg \cdot s^2}
]
This shows how ( G ) relates the gravitational force to the masses and the distance between them, playing a critical role in our understanding of gravitation in physics. The dimensions of ( G ) reflect its role in measuring how mass and distance influence gravitational attraction, encapsulating fundamental properties of gravity in the universe.