Newtons Law Of Gravitation

Let the density be for both the planets.
Given that
Now, mass of
similarly,
Escape velocity for a planet is given by

So,

Similarly,
Taking the ratio,

The necessary centripetal force required for a planet to move round the sun
= gravitational force exerted on it

or
as

Step 1: Force: F (1)/(Rⁿ)
Step 2: Equating centripetal force: (mv²)/(R) (1)/(Rⁿ) v R⁽¹⁻ⁿ⁾/⁽²⁾
Step 3: Time period: T = (2π R)/(v) R⁽ⁿ⁺¹⁾/⁽²⁾

Step 1: Escape velocity: vₑ = √((2GM)/(R))
Step 2: For same density, M R³.
Step 3: vₑ R (vA)/(vB) = (2R)/(R) = 2

Step 1: Shell theorem.
Spherical shell exerts zero force inside.
Step 2: Only central mass acts.
Effective mass inside = M/3
Step 3: Change in force.
Difference gives (2GMm)/(3R²)

Step 1: Gravitational force relation.
The acceleration due to gravity on the surface of the earth is given by: where is the gravitational constant, is the mass of the earth, and is the radius.

Step 2: Effect of change in radius.
If the radius decreases by 1%, then will increase, as gravity is inversely proportional to the square of the radius. The percentage increase in is approximately 2%.
Thus, the correct answer is (2).

Step 1: Understand the work done by the gravitational field.
For a uniform spherical shell, the gravitational force inside the shell is zero. Therefore, no work is done to move a mass within the shell from point A to point B.
Step 2: Conclusion.
The work done in this scenario is zero. Final Answer:

Step 1: Escape velocity from the surface of the earth is
Step 2: Given speed of the satellite is half of escape velocity:
Step 3: Orbital speed of a satellite in a circular orbit of radius is:
Step 4: Equating the two expressions for velocity:
Step 5: Height of the satellite above earth’s surface:

As we know, orbital speed, Time period Squaring both sides, .

Step 1: Radius of the circular orbit r = R + R = 2R
Step 2: Initial mechanical energy of the satellite at rest on the surface Eᵢ = -(GMm)/(R)
Step 3: Final mechanical energy of the satellite in a circular orbit Ef = -(GMm)/(2r) = -(GMm)/(4R)
Step 4: Minimum energy required to launch the satellite Δ E = Ef - Eᵢ = (-(GMm)/(4R)) - (-(GMm)/(R)) = (3GMm)/(4R)

Given each mass
and are placed in the following way
we know that
where mass , distance from origin
Now applying formula.
gravitational potential :




in G.P

Step 1: Potential due to one mass at distance r: V = -(G)/(r)
Step 2: Sum of reciprocals: (1 + (1)/(2) + (1)/(4) + ⋯ ) = 2
Step 3: Considering both sides: V = -2G × 2 = -4G
Step 4: Magnitude = 4G.

Step 1: Escape velocity: vₑ = √(2gR)
Step 2: Ratio: (vₑ)/(vm) = √((gₑ Rₑ)/(gm Rm))
Step 3: Substitute: (vₑ)/(vm) = √((gₑ Rₑ)/((gₑ)/(6)· (Rₑ)/(4))) = √(24) = dfrac√(6)2

Step 1: Velocity of a satellite in a circular orbit of radius r is v = √((GM)/(r)) Escape velocity from the earth’s surface is vₑ = √((2GM)/(Rₑ))
Step 2: Given that v = (vₑ)/(2) √((GM)/(r)) = (1)/(2)√((2GM)/(Rₑ))
Step 3: Squaring both sides, (GM)/(r) = (GM)/(2Rₑ) ⟹ r = 2Rₑ
Step 4: Height of the satellite above earth’s surface is h = r - Rₑ = 2Rₑ - Rₑ = Rₑ

Step 1: Understanding the Concept:
According to Shell Theorem, the gravitational force exerted by a uniform spherical shell on an object outside it is as if the entire mass were at the center. However, the force exerted by the shell on an object inside it is zero.

Step 2: Key Formula or Approach:
1. Force outside a sphere:
2. Force inside a shell:
3. Change in force

Step 3: Detailed Explanation:
1. Just outside the surface ( ): The man experiences gravity from the entire mass (shell + point mass). 2. Just inside the surface ( ): The man is now inside the spherical shell of mass . The shell exerts zero force on him. He only experiences the gravitational pull of the point mass located at the center. 3. Change in force:

Step 4: Final Answer
The change in the force of gravity experienced by the man is .