Mechanics (Part – 4)

G=6.67430×10⁻¹¹ m³⋅kg⁻¹⋅s⁻²

• F=Gm₁m₂/r²​​
  

Where:

• F is the gravitational force between two masses,
• m​₁ and m₂​ are the two masses.
• r is the distance between their centers, and G is the gravitational constant.

To find the acceleration due to gravity (g') at a distance of 2R from the surface of the Earth (where R is Earth's radius), we follow this process:

Step-by-step:

Total distance from the center of Earth
= Radius of Earth + 2R
= R + 2R = 3R

Formula:

g′=g/(r/R)^2​

Where:

• g = acceleration due to gravity on Earth's surface

• r = distance from the center of Earth = 3R

g′=g/(3)^2=g/9

Let the masses of the two objects be m1​ and m2​.

The gravitational force of attraction between them is given by Newton's law of gravitation:

F=Gm1​m2/r²

where G is the gravitational constant.

Now, if the distance between the two objects increases threefold, the new distance is $3r$.

The new gravitational force of attraction between them, F′, is:

F′=Gm₁m₂/(3r)²=Gm₁m₂/9r²=1/9(Gm₁m₂​​​​​​​​​​/r²)

$$F\prime =1/9(F)$$

$$F\prime =F/9$$​

The new acceleration due to gravity, g', is:

g=GM/R²,                            i.e., g∝1/R²      (if fixed)

​Therefore, if the radius of the Earth decreases and its mass remains the same, the value of "acceleration due to gravity" will increase.

• Mathematically, it's given by:
  F=Gm₁m₂/r²
  

Where:

• F is the gravitational force between the two bodies,
• G is the gravitational constant (6.674×10⁻¹¹ Nm²/kg²)
  
• m1 and m2 are the masses of the two bodies,
• r is the distance between the centers of the two masses.

• Its value is 6.67 x 10⁻¹¹ Newton m²/kg.
• Lord Cavendish first found its value in 1798.
• The value of G is the same in the whole universe.

Here's why:

Gravity inside the Earth decreases as you go deeper.

According to the shell theorem, when you're inside a spherical mass like the Earth, only the mass at a smaller radius than your position contributes to the gravitational pull. At the very center, you're surrounded equally in all directions, and all the gravitational forces cancel out.

So:

g(center)=0 m/s²

No net gravitational force acts on a mass placed at the Earth's center—it's in perfect gravitational equilibrium.

Sedimentation is the process of allowing particles in suspension in water to settle out of the suspension under the effect of gravity.

This is a common method used in water treatment to remove suspended solids before further purification steps like filtration or disinfection.

Therefore, the force that stops the vehicle when brakes are applied is called frictional force.