Step 1: Understanding the Concept:
Newton's second law of motion states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)).
We are given the accelerations produced by a constant force on two individual masses. We need to find the acceleration when this same force is applied to the sum of the two masses.
Step 2: Key Formula or Approach:
Using \( F = ma \), we can express mass as \( m = F/a \).
For the combined mass, the total mass is \( M = m_1 + m_2 \), and the new acceleration will be \( A = F/M \).
Step 3: Detailed Explanation:
Let the constant force be \( F \).
When force \( F \) acts on mass \( m_1 \), it produces acceleration \( A_1 \):
\[ F = m_1 A_1 \implies m_1 = \frac{F}{A_1} \]
When the same force \( F \) acts on mass \( m_2 \), it produces acceleration \( A_2 \):
\[ F = m_2 A_2 \implies m_2 = \frac{F}{A_2} \]
Now, the two masses are combined. The total combined mass \( M \) is:
\[ M = m_1 + m_2 \]
Substitute the expressions for \( m_1 \) and \( m_2 \):
\[ M = \frac{F}{A_1} + \frac{F}{A_2} \]
To add these fractions, find a common denominator:
\[ M = F \left( \frac{1}{A_1} + \frac{1}{A_2} \right) \]
\[ M = F \left( \frac{A_2 + A_1}{A_1 A_2} \right) \]
When the same force \( F \) acts on this combined mass \( M \), the new acceleration \( A \) produced is:
\[ A = \frac{F}{M} \]
Substitute the expression for \( M \):
\[ A = \frac{F}{F \left( \frac{A_1 + A_2}{A_1 A_2} \right)} \]
The force \( F \) cancels out from the numerator and the denominator:
\[ A = \frac{1}{\frac{A_1 + A_2}{A_1 A_2}} \]
\[ A = \frac{A_1 A_2}{A_1 + A_2} \]
Step 4: Final Answer:
The acceleration produced on the combined mass is \( \frac{A_1 A_2}{A_1 + A_2} \).