**There is a binary star system comprising of a pair of stars with masses denoted by $ m_1 $ and $ m_2 $ and centripetal acceleration denoted by $ a_1 $ and $ a_2 $. Both the stars, while attracting each other, circulate around a center of rotation of the combined system.**

This question aims to develop an understanding of **Newton’s laws of motion**, **centripetal force,** and **acceleration.**

According to Newton, a body’s **speed can’t be changed unless a force acts** on it to generate acceleration. Mathematically:

\[ F \ = \ m a \]

where $ F $ is the** force**, $ m $ is the **mass of the body** and $ a $ is the **acceleration.**

Whenever **bodies move in circular paths,** this type of motion is called **circulatory motion**. To perform or maintain a **circular motion,** a force is required that pulls the body towards the **axis of ****circulation**. This force is called the **centripetal force,** which is defined mathematically by:

\[ F \ = \ \dfrac{ m v^{ 2 } }{ r } \]

Where $ r $ is the **radius of the circular motion. **The **acceleration during circular motion** is also towards the center of the circulation, which is called **centripetal acceleration**. Comparing the above centripetal force equation with Newton’s second law, we can find the expression for the **centripetal acceleration**:

\[ a \ = \ \dfrac{ v^{ 2 } }{ r }\]

## Expert Answer

**Given that:**

\[ \text{ centripetal acceleration of star 1 } \ = \ a_1 \]

\[ \text{ centripetal acceleration of star 2 } \ = \ a_2 \]

\[ \text{ mass of star 1 } \ = \ m_1 \]

\[ \text{ mass of star 2 } \ = \ m_2 \]

**Assuming:**

\[ \text{ centripetal force of star 1 } \ = \ F_1 \]

\[ \text{ centripetal force of star 2 } \ = \ F_2 \]

**We can apply Newton’s law as follows:**

\[ F_1 \ = \ m_1 a_1 \]

\[ F_2 \ = \ m_2 a_2 \]

Since **both the stars exert equal and opposite force of gravitation** on each other, we can say that:

\[ \text{ centripetal force of star 1 } \ = \Â \text{ centripetal force of star 2 } \]

\[ F_1 \ = \ F_2 \]

\[ \Rightarrow m_1 a_1 \ = \ m_2 a_2 \]

**Solving for $ a_2 $:**

\[ \Rightarrow a_2 \ = \ \dfrac{ m_1 }{ m_2 } a_1 \]

## Numerical Result

\[ a_2 \ = \ \dfrac{ m_1 }{ m_2 } a_1 \]

## Example

If **mass of star 1 and star 2** are $ 20 \times 10^{ 27 } $ kg and $ 10 \times 10^{ 27 } $ kg respectively, and the **centripetal acceleration of star 1** is $ 10 \times 10^{ 6 } \ m/s^{2} $, then calculate the **centripetal acceleration of star 2.**

**Recall the equation:**

\[ a_2 \ = \ \dfrac{ m_1 }{ m_2 } a_1 \]

**Substituting values:**

\[ a_2 \ = \ \dfrac{ ( 20 \times 10^{ 27 } ) }{ ( 10 \times 10^{ 27 } ) } ( 10 \times 10^{ 6 } ) \]

\[ a_2 \ = \ 20 \times 10^{ 6 } \ m/s^{ 2 }\]