9. In Simple Harmonic Motion, no dissipative forces such as friction or viscosit
ID: 2040471 • Letter: 9
Question
9. In Simple Harmonic Motion, no dissipative forces such as friction or viscosity are present and so the system oscillates with maximum displacement each cycle to decrease. Consider a mass m connected to a spring (spring constant k) which is subject to a damping force that is proportional to, and opposite to, the velocity a constant maximum displacement. In reality, dissipative forces cause the of the mass ; specifically, damping =-bi, where b is the damping coefficient that measures the strength of the damping. For weak damping the mass oscillates but with time: with a maximum displacement () each cycle that decreases exponentially where A is the maximum displacement at time tO. A 300 gram mass is attached to a spring and oscillates as a damped mass/spring system. A graph of displacement as a function of time is shown below. Assume the magnitude of the damping force is proportional to the speed of the mass. a) What is the period of the motion? b) What is the damping coefficient? 10.18 kg/s) 2.0 1.0 0.0 1.0 Time (s) -2.0 Page 219 of 281 PHYS-111 College Physics I Activities ManualExplanation / Answer
The damped harmonic oscillation is basically,
xmax = A e-(b/2m)t cos (?t)....(1), where,
xmax = maximum displacement,
A = maximum displacement at time t = 0,
b = damping co-efficient
m = mass = 300 g (as given in problem)
?t = phase angle
As per given problem and comparing with (1), we have,
cos (?t) = 1 cos (2n?)
Or ?t = 2n?, where n = 1, 2, 3, 4,....
a) Thus, the period of the motion "t" = (2n?) /? where n = 1, 2, 3, 4,....
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