Final answer:
The number of periods for a mass on a spring can be calculated using the time period formula T = total time / number of cycles, resulting in T for given example as 0.6 seconds per cycle. The frequency is the reciprocal of this period. The galactic year is around 225-250 million Earth years for our Sun, but requires specific galactic velocities and distances for an accurate calculation.
Step-by-step explanation:
Calculating the Number of Periods
When calculating the number of periods for an oscillating object, such as a mass on a spring, we use the formula for the time period (T) which is T = total time / number of cycles. In the example of a mass attached to a spring completing 50 full cycles in 30 seconds, the time period T would be 30 s / 50 cycles = 0.6 seconds per cycle. The frequency (f), which is the number of cycles per second, would be the reciprocal of the period, f = 1/T, giving us a frequency of 1/0.6 s ≈ 1.67 Hz.
For an object traveling around the Sun with a semimajor axis of 50 AU, we would use Kepler's third law to calculate the orbital period, which states that the orbital period squared is proportional to the semimajor axis cubed (T² ≈ a³). However, without additional context or the constant of proportionality specific to the Sun's gravitational influence, the calculation cannot be completed here.
For calculating the Sun's period, the 'galactic year,' one would require the speed of the orbit and the distance of the entire orbit to calculate it correctly. For an Earth observer, the galactic year is approximately 225-250 million Earth years, but without the specific numbers associated with the Sun's galactic orbit, any present calculation here would be speculative.