Question

There are two identical spherical solid balls of iron alloy touching each other. What would be the gravitational force of attraction between them if the density of the materials of the ball is

        (3/2π)x10^5(kg/m^3)
      

and radius 1 m.

        G=6.67×10−11Nm2/kg2
      

• A. 0.667 N
• B. 0.0667 N
• C. 66.7 N
• D. 6.67 N

Answer 1

Answer:

A. 0.667 N

Gravitational Force Between Two Identical Spheres

Understanding the Concept

• The gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is described by Newton's Law of Universal Gravitation.

• Formula: F = G * (m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.

Given Information Analysis

• Two identical spherical solid balls.

• Density of the material (ρ) = (3/2π) x 10^5 kg/m³.

• Radius of each ball (R) = 1 m.

• Gravitational constant (G) = 6.67 x 10⁻¹¹ Nm²/kg².

• The balls are touching each other. This means the distance between their centers (r) is equal to the sum of their radii, so r = R + R = 2R.

Calculations

1. Calculate the volume of one sphere:
   V = (4/3)πR³
   V = (4/3)π(1 m)³ = (4/3)π m³

2. Calculate the mass of one sphere:
   Mass (m) = Density (ρ) x Volume (V)
   m = [(3/2π) x 10^5 kg/m³] * [(4/3)π m³]
   m = (3/2π) * (4/3)π * 10^5 kg
   m = (3 * 4 * π) / (2 * π * 3) * 10^5 kg
   m = 6π / 6π * 10^5 kg
   m = 2 x 10^5 kg

3. Determine the distance between the centers of the two spheres:
   Since the balls are touching, the distance between their centers (r) is the sum of their radii.
   r = R + R = 1 m + 1 m = 2 m

4. Calculate the gravitational force:
   F = G * (m * m) / r² (since m1 = m2 = m)
   F = G * m² / r²
   F = (6.67 x 10⁻¹¹ Nm²/kg²) * (2 x 10^5 kg)² / (2 m)²
   F = (6.67 x 10⁻¹¹ Nm²/kg²) * (4 x 10^10 kg²) / (4 m²)
   F = 6.67 x 10⁻¹¹ x 4 x 10^10 / 4 N

5. Simplify the expression:
   F = 6.67 x 10⁻¹ N
   F = 0.667 N