A force produces an acceleration of `a_(1)` in a body and the same force produces an acceleration of `a_(2)` in another body. If the two bodies are combined and the same force is applied on the combination, the acceleration produced in it is

Options

A. `a_(1) + a_(2)`
B. `(a_(1) + a_(2))/(a_(1)a_(2))`
C. `(a_(1)a_(2))/(a_(1) + a_(2))`
D. `sqrt(a_(1)a_(2))`

Answer

To solve the problem, we need to find the acceleration produced when two bodies are combined and the same force is applied to them. Let's break it down step by step.

### Step 1: Understand the relationship between force, mass, and acceleration
According to Newton's second law of motion, the force acting on an object is equal to the mass of the object multiplied by its acceleration. This can be expressed as:
\[ F = m \cdot a \]

### Step 2: Define the forces acting on the individual bodies
Let:
- \( m_1 \) be the mass of the first body,
- \( a_1 \) be the acceleration of the first body when force \( F \) is applied,
- \( m_2 \) be the mass of the second body,
- \( a_2 \) be the acceleration of the second body when the same force \( F \) is applied.

From the definition of force, we can write:
\[ F = m_1 \cdot a_1 \quad (1) \]
\[ F = m_2 \cdot a_2 \quad (2) \]

### Step 3: Express the masses in terms of force and acceleration
From equations (1) and (2), we can express the masses as:
\[ m_1 = \frac{F}{a_1} \quad (3) \]
\[ m_2 = \frac{F}{a_2} \quad (4) \]

### Step 4: Combine the masses of the two bodies
When the two bodies are combined, the total mass \( M \) is:
\[ M = m_1 + m_2 \]
Substituting equations (3) and (4) into this, we get:
\[ M = \frac{F}{a_1} + \frac{F}{a_2} \]

### Step 5: Factor out the force
Factoring out \( F \), we have:
\[ M = F \left( \frac{1}{a_1} + \frac{1}{a_2} \right) \]

### Step 6: Apply the same force to the combined mass
Now, when the same force \( F \) is applied to the combined mass \( M \), the acceleration \( a \) produced can be expressed as:
\[ F = M \cdot a \]
Substituting for \( M \):
\[ F = \left( \frac{F}{a_1} + \frac{F}{a_2} \right) \cdot a \]

### Step 7: Simplify the equation
Dividing both sides by \( F \) (assuming \( F \neq 0 \)):
\[ 1 = \left( \frac{1}{a_1} + \frac{1}{a_2} \right) \cdot a \]

### Step 8: Solve for acceleration \( a \)
Rearranging the equation gives:
\[ a = \frac{1}{\left( \frac{1}{a_1} + \frac{1}{a_2} \right)} \]
This can be simplified to:
\[ a = \frac{a_1 \cdot a_2}{a_1 + a_2} \]

### Final Answer
Thus, the acceleration produced in the combined body when the same force is applied is:
\[ a = \frac{a_1 \cdot a_2}{a_1 + a_2} \]

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