Suppose that a spring with a larger spring constant was used in this same apparatus. If a given mass were rotated at the same radius at which it had been rotated with the original spring, would the new period of rotation using the new spring be greater or less than the period of rotation using the original spring

The new period of rotation using the new spring would be less than the period of rotation using the original spring

Explanation:

Generally the  period of rotation of the mass is mathematically represented as

Here I is the moment of inertia of the mass about the rotation axis and  k is the spring constant

Now looking at the equation we can tell that T is inversely proportional to the square root of the spring constant which means that for a larger spring constant the time period would be lesser