Two masses A & B each of `5 kg` are suspended by a light inextensible string passing over a smooth massless pulley such that mass A rest on smooth table & B is held at the position shown. Mass B is now gently lifted up to the pulley and allowed to fall from rest. Determine up to what height will A rise for the ensuing motion.

Explanation

To solve the problem step by step, we will analyze the motion of the two masses A and B.

### Step 1: Understand the System
We have two masses, A and B, each of mass 5 kg. Mass A is resting on a smooth table, and mass B is held at a height and then allowed to fall. When mass B falls, it will cause mass A to rise due to the tension in the string.

### Step 2: Analyze the Motion of Mass B
When mass B is released from rest, it will fall under the influence of gravity. The height from which mass B falls is the same height that mass A will rise. We can denote the height that mass B falls (and thus the height that mass A rises) as \( h \).

### Step 3: Apply Energy Conservation
Since there are no non-conservative forces doing work (the pulley is massless and frictionless), we can use the principle of conservation of energy. The potential energy lost by mass B will be equal to the potential energy gained by mass A.

The potential energy lost by mass B when it falls a height \( h \) is:
\[ PE_B = m_B \cdot g \cdot h = 5 \cdot g \cdot h \]

The potential energy gained by mass A when it rises a height \( h \) is:
\[ PE_A = m_A \cdot g \cdot h = 5 \cdot g \cdot h \]

### Step 4: Set the Energies Equal
Since the potential energy lost by mass B is equal to the potential energy gained by mass A, we can set these two equations equal to each other:
\[ 5 \cdot g \cdot h = 5 \cdot g \cdot h \]
This is trivially true, indicating that the system conserves energy.

### Step 5: Determine the Height
To find the height \( h \) that mass A rises, we need to consider the distance that mass B falls. If mass B is lifted to the pulley and then released, it will fall a distance equal to the length of the string that was initially held.

Assuming mass B falls a distance \( h \), mass A will rise the same distance \( h \). Thus, if mass B falls from a height of 1 meter (for example), mass A will rise by the same height.

### Conclusion
If mass B is released from a height of 1 meter, mass A will rise to a height of 1 meter as well. Therefore, the height \( h \) that mass A rises is:
\[ h = 1 \text{ meter} \]