A

Step 1: Consider the portion of the chain that is falling. The entire chain has a mass \( M \) and length \( \lambda \). When the top end of the chain falls by a distance \( \frac{\lambda}{8} \), the length of the chain above the table will be \( \lambda - \frac{\lambda}{8} = \frac{7\lambda}{8} \).
Step 2: The mass of the portion of the chain above the table can be calculated as follows: \( m_{above} = M \cdot \frac{(7\lambda/8)}{\lambda} = \frac{7M}{8} \).
Step 3: The potential energy lost by the chain when it falls this distance is converted into kinetic energy. This kinetic energy will be equal to the kinetic energy of the mass portion that falls.
The potential energy (PE) lost is given by \( PE = m_{above} imes g imes h = \frac{7M}{8} \cdot g \cdot \frac{\lambda}{8} \).
Step 4: The kinetic energy (KE) at the point of falling is given by \( KE = \frac{1}{2} mv^2 \). Setting these equal: \( \frac{7M}{8} \cdot g \cdot \frac{\lambda}{8} = \frac{1}{2} \cdot \frac{7M}{8} \cdot v^2 \).
Step 5: From the equation, we can find \( v \): \( g \cdot \frac{\lambda}{8} = \frac{1}{2} v^2 \Rightarrow v^2 = 2g \cdot \frac{\lambda}{8} \Rightarrow v = \sqrt{g \cdot \frac{\lambda}{4}} \).
Step 6: The momentum (p) of the mass above the table after falling is given by \( p = m_{above} \cdot v = \frac{7M}{8} \cdot \sqrt{g \cdot \frac{\lambda}{4}} \).
Therefore, the momentum of the portion of the chain above the table after falling a distance \( \frac{\lambda}{8} \) is expressed as \( p = \frac{7M}{8} \cdot \sqrt{g \cdot \frac{\lambda}{4}} \).
Thus, we conclude that this momentum is a factor of \( \frac{7}{8} \) of the total momentum when the height change corresponds to the above expression.