Numerical Problems Based on Gravitational Constant (G) for JEE and CBSE Preparations:

Assistant:

1. Cavendish’s Experiment:

Cavendish’s experiment demonstrates the gravitational force between two masses. To calculate the value of G, you need the dimensions and observations from the experimental setup, including the masses of the spheres, the distance between them, and the angle of deflection. Use the formula:
G = (F * L²)/(m₁ * m₂ * r²)
Where F is the gravitational force, L is the length of the torsion fiber, m₁ and m₂ are the masses of the spheres, and r is the distance between their centers.

2. Measurement of G Using Pendulums:

Hang two identical pendulums side by side, one with a small lead sphere and the other with a larger lead sphere. Measure their oscillation periods (T₁) for the smaller sphere and (T₂) for the larger sphere. Use the formula: G = (4π²(L/g) * (T₁²-T₂²))/(D * ρ * r)
Where L is the length of the pendulums, g is the acceleration due to gravity, D is the difference in diameters of the spheres, ρ is the density of the lead, and r is the radius of the spheres.

3. Orbital Motion and G:

Use the orbital period (T = 365.25 days) and the radius of Earth’s orbit around the Sun (r = 1.496 × 10^11 m).
Substitute these values into the formula: G = (4π²r³/T²)

4. Satellite and G:

Use the given mass of the satellite (m), the mass of the planet (M), and the distance between them (r).
Substitute these values into the universal gravitation formula: *G = F r²/(m₁ * m₂)
Where F is the gravitational force between the satellite and the planet.**

5. Free Fall and G:

Measure the height (h) from which the object is dropped, and use a stopwatch to accurately determine the time of flight (t).
Calculate the acceleration due to gravity (g) using: g = 2h/t²
Rearrange the universal gravitation formula to solve for G: **G = g * R²/M
Where R is the radius of the Planet.**

6. Earth’s Density and G:

Rearrange the formula: M = (4/3)πR³ρ
Where M is the mass of the Earth, R is the radius of the Earth, and ρ is its average density.
Solve for ρ and substitute the known values of G, M, and R.

7. Gravitational Potential at the Surface of Mars:

Use the formula for gravitational potential: Φ = -(GM)/R
Where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.

8. Escape Velocity:

Use the formula for escape velocity: V(escape) = [2GM/R]^(1/2)
Where G is the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.