Question

Block and Spring in SHM with damping

A block of mass 8.50 g on the end of spring undergoes simple harmonic motion with a frequency of 9.00 Hz.

What is the spring constant of the spring?

If the motion of the mass has an initial amplitude of 6.00 cm what is its maximum speed?.

The amplitude decreases to 1.335 cm in 1.78 s, what is the damping constant for the system?

Answer 1

Angular velocity:

$w=2\pi f=\sqrt{\frac{k}{m}}$

$k=(2\pi *9)^{2}0.0085\rightarrow \boldsymbol{k=271.81\frac{N}{m}}$

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Energy:

$\frac{1}{2}kA^{2}=\frac{1}{2}mv^{2}\rightarrow v^{2}=\frac{271.81(0.06)^{2}}{0.0085}$

$\boldsymbol{v=115.12\frac{m}{s}}$

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Generel solution to problem:

$x(t)=Ae^{\frac{-b}{2m}t}\sin (\sqrt{\frac{k}{m}}t)$

for t=1.78

$\frac{0.01335}{0.06\sin (56.55*1.78)}=e^{\frac{-b}{2*0.0085}1.78}$

$e^{\frac{b}{2*0.0085}1.78}=4.42$

Apply log to both sides:

$\frac{b}{2*0.0085}1.78=ln(4.42)$

$\boldsymbol{b=0.014}$