The gravitational force between two masses, P and Q, is 10N, find the new value of the force if both masses are doubled

1. Analyze the question

We are given the initial gravitational force between two masses and asked to calculate the new force when both masses are doubled. This requires an understanding of Newton’s Law of Universal Gravitation.

2. Explain the concept

Newton’s Law of Universal Gravitation states that the force ($F$) between two point masses ($m_1$ and $m_2$) separated by a distance ($r$) is given by the formula:

Where $G$ is the gravitational constant.

From this formula, we can see that the force is directly proportional to the product of the masses ($F \propto m_1 m_2$). If the masses change, the force changes accordingly.

3. Show the solution

• Initial state: $F_1 = 10\text{N} = \frac{G m_P m_Q}{r^2}$
• New state: Both masses are doubled. Let the new masses be $m_P' = 2m_P$ and $m_Q' = 2m_Q$. The distance $r$ remains unchanged.
• New force ($F_2$): $$F_2 = \frac{G (2m_P) (2m_Q)}{r^2}$$ $$F_2 = \frac{G \cdot 4 \cdot m_P m_Q}{r^2}$$ $$F_2 = 4 \times \left( \frac{G m_P m_Q}{r^2} \right)$$

Since we know that $\frac{G m_P m_Q}{r^2} = 10\text{N}$, we substitute this into the equation:

4. Identify the answer

The calculated force is 40.0N, which matches option A.

5. Tip

When dealing with proportionality questions in Physics, you don't always need to plug in the value of $G$. Just look at the "multiplier." If both masses double, the product of the masses increases by $2 \times 2 = 4$. Therefore, the force simply increases by a factor of 4. If the distance had also changed, you would apply the square of that change to the denominator!