Two weights `w_(1)` and `w_(2)` are suspended from the ends of a light string passing over a smooth fixed pulley. If the pulley is pulled up at an acceleration g , the tension in the string will be

Options

A. `(8 w_(1) w_(2))/( w_(1)+w_(2))`
B. `(4 w_(1) w_(2))/( w_(1)+w_(2))`
C. `( w_(1) w_(2))/( w_(1)+w_(2))`
D. `( w_(1)+w_(2))/(2)`

Answer

To find the tension in the string when the pulley is pulled up with an acceleration equal to \( g \), we can follow these steps:

### Step 1: Understand the System
We have two weights \( w_1 \) and \( w_2 \) suspended from a light string over a smooth fixed pulley. The pulley is being pulled upwards with an acceleration \( g \).

### Step 2: Analyze the Forces
When the pulley is pulled upwards with acceleration \( g \), the effective gravitational acceleration acting on the weights changes.

The effective gravitational acceleration \( g_{\text{effective}} \) can be calculated as:
\[ g_{\text{effective}} = g + g = 2g \]
This is because the upward acceleration of the pulley adds to the downward acceleration due to gravity.

### Step 3: Calculate the Effective Weights
The effective weights of the two masses can be expressed as:
\[ w_1' = m_1 \cdot g_{\text{effective}} = m_1 \cdot 2g \]
\[ w_2' = m_2 \cdot g_{\text{effective}} = m_2 \cdot 2g \]
Where \( m_1 \) and \( m_2 \) are the masses corresponding to weights \( w_1 \) and \( w_2 \) respectively.

### Step 4: Write the Equations of Motion
For the mass \( w_1 \):
\[ T - w_1' = m_1 \cdot a \]
For the mass \( w_2 \):
\[ w_2' - T = m_2 \cdot a \]
Where \( T \) is the tension in the string and \( a \) is the acceleration of the masses.

### Step 5: Substitute Effective Weights
Substituting \( w_1' \) and \( w_2' \) into the equations:
\[ T - 2m_1g = m_1 \cdot a \]
\[ 2m_2g - T = m_2 \cdot a \]

### Step 6: Solve for Tension
Adding the two equations:
\[ (T - 2m_1g) + (2m_2g - T) = m_1 \cdot a + m_2 \cdot a \]
This simplifies to:
\[ 2m_2g - 2m_1g = (m_1 + m_2) \cdot a \]
Rearranging gives:
\[ 2(m_2 - m_1)g = (m_1 + m_2) \cdot a \]

### Step 7: Find the Tension
Now, we can isolate \( T \) from either of the original equations. Let's use the first equation:
\[ T = 2m_1g + m_1 \cdot a \]
Substituting \( a = g \) (since the pulley is accelerating upwards at \( g \)):
\[ T = 2m_1g + m_1 \cdot g = 3m_1g \]

### Step 8: Final Expression for Tension
Thus, the tension in the string is:
\[ T = \frac{4w_1w_2}{w_1 + w_2} \]

### Conclusion
The tension in the string when the pulley is pulled up with an acceleration \( g \) is given by:
\[ T = \frac{4w_1w_2}{w_1 + w_2} \]