Gravitation

Derivation:
The formula for gravitational force is given by 2F=G⋅r2m1​⋅m2​​.
Alternatively, we can express G using dimensional analysis:
Starting with 2F=G⋅r2m1​⋅m2​​, we rearrange it to solve for G:
2G=m1​⋅m2​F⋅r2​ . . . . . (1)
Here, G represents the Universal Gravitational Constant.
Now, let's analyze the dimensions of the involved quantities:
Mass has dimensions: [M] Radius has dimensions: [L] Force has dimensions: [M⋅L⋅T−2]
Substituting these dimensions into equation (1), we get:
G=[M]⋅[M][M⋅L⋅T−2]⋅[L]2​
Simplifying further:
G=[M−1⋅L3⋅T−2]
Hence, the dimension of the Universal Gravitational Constant (G) is [M−1⋅L3⋅T−2].
M^-1L³T^-2

Step 1: Formula for escape velocity
The escape velocity from a planet is given by: where: - is the gravitational constant
- is the mass of the planet
- is the radius of the planet

Step 2: Original planet’s escape velocity
Let the original planet have mass and radius and radius . Its escape velocity is:
Step 2: New planet’s parameters
The new planet has: - Mass =
- Radius =

Step 3: Calculate new escape velocity
Substitute and into the escape velocity formula: The new escape velocity , so it remains unchanged.

Step 4: Alternative approach
Notice that escape velocity depends on the ratio : For the new planet: Since the ratio is unchanged, the escape velocity remains . Thus, the correct answer is option (1) .