Question

6. -0 points My Notes O Ask Your Teache A 10.1 kg object oscillates at the end of a vertical spring that has a spring constant of 2.20 x 104 N/m. The effect of air resistance is represented by the damping coefficient b = 3.00 N-s/m (a) Calculate the frequency of the dampened oscillation. H2 (b) By what percentage does the amplitude of the oscillation decrease in each cycle? 1% 7. -0 points SerPSE9 15.P063 MI.FB My Notes Ask Your Teache The free-fall acceleration on Mars is 3.7 m/s2 a) What length of pendulum has a period of 2.2 s on Earth? cm (b) What length of pendulum would have a 2.2-s period on Mars? cm An object is suspended from a spring with force constant 10 N/m. c) Find the mass suspended from this spring that would result in a period of 2.2s on Earth kg (d) Find the mass suspended from this spring that would result in a period of 2.2 s on Mars kg

please give me answers to all the questions and i would really appreciate that

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Answer 1

Given m = 10.1 kg, k = 2.2 times 10^4 N/m, b = 3.00 N-s/m Now angular frequency of damped oscillator is w = squareroot (k/m) - (b/2m)^2 = squareroot 2.2 times 10^4/10.1 - 3/2 times (10.1) = squareroot 21.7 times 10^2 - 0.1485 = squareroot 2178 - 0.1485 w = 46.66 sec^-1 Now, frequency f = w/2 pi = 46.66/2 times 3.14 f = 7.43 Hz The equation of the damped oscillator is x = A e^-(b/2m)t cos (wt + phi) For one cycle, let the initial amplitude be A_0 decrease in the amplitude A_0 - A_0 e^-(b/2m)t Fractional decrease A_0 - A_0 e^-(b/2m) t/A \r\n Here time t = 2 pi/w then (A_0 - A_0 e^-(b/2m) (2 pi/w)/A) = 1 - e^-b pi/m w = 1 - e^-3 times 3.14 times 10.1 times 46.66 = 0.0197 Now percentage = 1.97 % The free fall acceleration on mars is (g prime) = 3.7 m/sec^2 The free fall acceleration on earth is (g) = 9.8 m./sec^2