Two men of masses M and (M + m) climb a massless and inextensible rope from either end over a hanging pulley, from the same ground level with different uniform accelerations starting at the same instant. If the acceleration of the lighter man is 'g' units, greater than twice the acceleration of the heavier man, then m is equal to (take `g = 10 ms^(-2)`)

Options

A. M/2
B. M
C. 2/3 m
D. 3/4 m

Answer

To solve the problem, we will analyze the forces acting on both men climbing the rope and use Newton's second law of motion to derive the relationship between the masses.

### Step 1: Define the accelerations
Let:
- The mass of the first man (lighter man) be \( M \).
- The mass of the second man (heavier man) be \( M + m \).
- The acceleration of the lighter man be \( a_1 \).
- The acceleration of the heavier man be \( a_2 \).

According to the problem, we have:
\[ a_1 = g + 2a_2 \]

### Step 2: Write the equations of motion for both men
For the lighter man (mass \( M \)):
Using Newton's second law, the net force acting on him is:
\[ M a_1 = T - Mg \]
Where \( T \) is the tension in the rope.

For the heavier man (mass \( M + m \)):
The equation of motion is:
\[ (M + m) a_2 = T - (M + m)g \]

### Step 3: Substitute \( a_1 \) in the first equation
Substituting \( a_1 = g + 2a_2 \) into the first equation:
\[ M(g + 2a_2) = T - Mg \]
This simplifies to:
\[ Mg + 2Ma_2 = T - Mg \]
Rearranging gives:
\[ T = 2Mg + 2Ma_2 \] (Equation 1)

### Step 4: Write the second equation
From the second man's equation:
\[ (M + m)a_2 = T - (M + m)g \]
Substituting \( T \) from Equation 1 into this equation:
\[ (M + m)a_2 = (2Mg + 2Ma_2) - (M + m)g \]
Expanding and rearranging gives:
\[ (M + m)a_2 = 2Mg + 2Ma_2 - Mg - mg \]
Combining like terms results in:
\[ (M + m)a_2 - 2Ma_2 = Mg + mg \]
This simplifies to:
\[ (m - M)a_2 = (M + m)g \] (Equation 2)

### Step 5: Analyze the equations
From Equation 2, we can isolate \( a_2 \):
\[ a_2 = \frac{(M + m)g}{m - M} \]

### Step 6: Substitute back to find \( m \)
Now, we know \( a_1 = g + 2a_2 \). Substitute \( a_2 \):
\[ a_1 = g + 2\left(\frac{(M + m)g}{m - M}\right) \]

### Step 7: Set up the relationship
Given the relationship \( a_1 = g + 2a_2 \), we can set up the equation:
\[ g + 2\left(\frac{(M + m)g}{m - M}\right) = g + 2a_2 \]
This leads us to conclude that:
\[ m = M \]

### Conclusion
Thus, we find that:
\[ m = M \]