You can try to solve this problem using law of conservation of energy and using law of conservation of angular momentum and you will get different answers.
The thing is that you can't use conservation of energy law in this case. At the moment when the string became taut some kind of inelastic impact would happen and some portion of energy would lost. Imagine, that the string is made of elastic rubber so that no energy would be lost. But in this case there will be some oscillations. In case the string is rigid these oscillations will fade out quickly, but some energy will be lost in the process.
Use the law of conservation of angular momentum.
Since it's a homework I will not provide a full solution. Just main steps.
In the beginning the block just falls. You can find it's speed $V_1$ at the moment just before the string gets taut.
Total angular momentum of the system at that moment would be $-m V_1 R$ - we calculate the momentum around the center of disk.
After the string is taut the speed of the block will be some some $V_2$, the angular velocity of disk $\omega$, and $V_2 = \omega * R$.
Now you calculate the total momentum of disk and block.
These two calculated angular momentum should be equal. And this would be the first equation on your picture.