Final answer:
The correct answer is b) T/√2. The time period of a simple harmonic oscillator depends on the mass and spring constant. Cutting a spring and arranging the pieces in parallel increases the effective spring constant, thus the new period is T/√2.
Step-by-step explanation:
The time period of an oscillator in simple harmonic motion is determined by the mass attached to it and the force constant of the spring.
The equation for the period T of a mass m attached to a spring with force constant k is T = 2π√(m/k).
When the spring is cut into two and arranged in parallel, the effective spring constant keff becomes the sum of the spring constants of each half because the spring constants are additive in parallel.
Therefore, if each half has a spring constant of k/2, the effective spring constant is keff = k/2 + k/2 = k.
However, when the original spring is cut in half, each half would actually have a spring constant 2k, because spring constant is inversely proportional to the length of the spring (assuming they are identical in material and cross-section). In parallel, their combined spring constant would be keff = 2k + 2k = 4k.
This increase in spring constant will affect the period of oscillation.
The new time period Tnew for the mass m with the new combined spring constant keff is calculated as follows:
Tnew = 2π√(m/4k)
= (1/2)√(m/k)
= T/√2,
since the original period T = 2π√(m/k).
Therefore, the correct answer is b) T/√2.