Chapter 14, Problem 14.90P

DATA You hang various masses m from the end of a vertical, 0.250-kg spring that obeys Hooke’s law and is tapered, which means the diameter changes along the length of the spring. Since the mass of the spring is not negligible, you must replace m in the equation $T=2\pi \sqrt{m/k}$ with m + m_eff, where m_eff is the effective mass of the oscillating spring. (See Challenge Problem 14.93.) You vary the mass m and measure the time for 10 complete oscillations, obtaining these data:

(a) Graph the square of the period T versus the mass suspended from the spring, and find the straight line of best fit. (b) From the slope of that line, determine the force constant of the spring. (c) From the vertical intercept of the line, determine the spring’s effective mass. (d) What fraction is m_eff of the spring’s mass? (c) It a 0.450-kg mass oscillates on the end of the spring, find its period, frequency, and angular frequency.