Solution

Statement

$2.3.7.$ A window blind of mass $m = 1\text{ kg}$ and length $L = 2\text{ m}$ is rolled up into a thin roller above the window. What is the minimum work spent on this? Neglect friction.

Solution

To roll the blind into a thin roller at the top, we need to raise its center of mass. Initially, for a uniform rectangular blind of length $L$, the center of mass is located precisely in the middle, at a distance of $\frac{L}{2}$ from the top.

Therefore, the minimum required work is equal to the change in the potential energy of the blind's center of mass:
$$A = mg\frac{L}{2}$$

Substituting the given values:
$$A = 1 \cdot 10 \cdot \frac{2}{2} = 10\text{ J}$$