Two rings A and B are parallel to each other and their centres are placed 10 m from each other. Ring A has radius 1m and mass 1kg, Ring B has radius 2m and mass 2kg . find where the 1kg has to be placed so that it does not move towards 2kg(in between those 10 m).

To determine where the 1 kg mass should be placed between the two rings so that it remains stationary and does not move towards the 2 kg mass, we need to consider the gravitational forces acting on the mass due to both rings. The key is to find a point where these forces balance each other out.

Understanding Gravitational Forces

The gravitational force exerted by a ring on a point mass located along its axis can be calculated using the formula:

F = (G * M * m) / r²

Where:

• F is the gravitational force.
• G is the gravitational constant (approximately 6.674 × 10⁻¹¹ N(m/kg)²).
• M is the mass of the ring.
• m is the mass of the object being influenced (in this case, 1 kg).
• r is the distance from the center of the ring to the point mass.

Setting Up the Problem

Let's denote the distance from the center of ring A to the 1 kg mass as x. Consequently, the distance from the center of ring B to the 1 kg mass will be (10 - x), since the total distance between the two rings is 10 m.

Calculating Forces

The gravitational force exerted by ring A on the 1 kg mass is:

F_A = (G * 1 kg * 1 kg) / x²

The gravitational force exerted by ring B on the 1 kg mass is:

F_B = (G * 2 kg * 1 kg) / (10 - x)²

Finding the Balance Point

For the 1 kg mass to remain stationary, the forces must be equal:

F_A = F_B

This leads us to the equation:

(G * 1 kg * 1 kg) / x² = (G * 2 kg * 1 kg) / (10 - x)²

We can simplify this by canceling out the common terms:

1 / x² = 2 / (10 - x)²

Cross-Multiplying

Cross-multiplying gives us:

(10 - x)² = 2x²

Expanding the left side:

100 - 20x + x² = 2x²

Rearranging the equation:

0 = x² + 20x - 100

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Here, a = 1, b = 20, and c = -100.

Calculating the discriminant:

b² - 4ac = 20² - 4(1)(-100) = 400 + 400 = 800

Now substituting into the quadratic formula:

x = (-20 ± √800) / 2

Calculating the square root of 800 gives us approximately 28.28:

x = (-20 ± 28.28) / 2

This results in two potential solutions:

• x ≈ 4.14 m (valid, between the rings)
• x ≈ -24.14 m (not valid, outside the range)

Final Placement

Thus, the 1 kg mass should be placed approximately 4.14 meters from the center of ring A to ensure that it remains stationary and does not move towards the 2 kg mass in ring B. This point is where the gravitational forces from both rings balance each other out perfectly.