Two masses m and m' are tied with a thread passing over a pulley, m' is on a frictionless horizontal surface and m is hanging freely. If acceleration due to gravity is g, the acceleration of m' in this arrangment will be `:`

Options

A. g
B. `mg //( m + m ')`
C. `mg //m'`
D. `mg // ( m - m')`

Answer

To solve the problem, we will analyze the forces acting on the two masses and apply Newton's second law of motion.

### Step-by-Step Solution:

1. **Identify the System**: We have two masses, \( m \) (hanging mass) and \( m' \) (mass on the horizontal surface). The mass \( m' \) is on a frictionless surface, and \( m \) is hanging freely.

2. **Determine the Forces**: The only force acting on the hanging mass \( m \) is its weight, which is given by \( F = mg \), where \( g \) is the acceleration due to gravity.

3. **Apply Newton's Second Law**: According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)).

4. **Calculate the Net Force**: The net force acting on the system is the weight of the hanging mass \( m \), which causes both masses to accelerate. Thus, the net force \( F_{net} = mg \).

5. **Total Mass of the System**: The total mass being accelerated is the sum of both masses, which is \( m + m' \).

6. **Set Up the Equation for Acceleration**: Using the formula for acceleration \( a = \frac{F_{net}}{m_{total}} \), we can substitute the values:
\[ a = \frac{mg}{m + m'} \]

7. **Final Expression for Acceleration**: Therefore, the acceleration of mass \( m' \) (and also mass \( m \) since they are connected) is given by:
\[ a = \frac{mg}{m + m'} \]

### Final Answer:
The acceleration of mass \( m' \) is \( \frac{mg}{m + m'} \).