Answer :
The spring constant for a mass of 51.7 grams on a spring that undergoes 21 cycles with a small amplitude in 3.42 seconds is 76.8 N/m.
The value of k for a mass on a spring can be determined using the formula T=2π√(m/k), where T is the period of oscillation, m is the mass, and k is the spring constant. In this problem, we know that the mass is 51.7 grams and that 21 cycles took 3.42 seconds, which means that the period of oscillation is T=3.42/21=0.163 seconds. Since the amplitude is small, we can assume that the motion is simple harmonic, which means that T=2π√(m/k) can be used. Rearranging this formula gives k=m(2π/T)^2, which gives k=51.7(2π/0.163)^2=76.8 N/m.
This value was calculated using the formula k=m(2π/T)^2, where m is the mass and T is the period of oscillation.
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