Question

The 11 g head of a bobble-head doll oscillates in SHM at a frequency of 5.0 Hz . What is the spring constant of the spring on which the head is mounted? Suppose the head is pushed ‎2.0 cm against the spring, then released. The amplitude of the head's oscillations decreases to 0.5cm in 4.0s. What is the head's damping constant? Ive asked this question multiple times and nobody can get it right.

Answer 1

A)

Given

Mass of the head, m = 0.011 kg

Frequency of the oscillation, f = 5.0 Hz

From the relation, f = (1/2π) √k/m

5 = 1/(2pi)*sqrt(k/0.011)

from here K =10.8566 N/m

The maximum speed , v_max = x_mω

The amplitude ,  x_m     = 0.02 m

Angular frequency , ω = √k / m = sqrt( 10.8566/0.011) = 31.4159774 rad/s

v_max = x_mω = 0.02* 31.4159774 = 0.628319548 m/s

B)

The oscillations decreases to 0.5 cm in t = 4.0s, the damping constant b is

    calculated from the relation

                                  x(t)   =    x_m e ^-bt/2m

0.005  m = (0.02 m) e ^-bt/2m  ​

-bt / 2m = ln(0.25)

= -b*4 /( 2*0.011) = -1.3862

  b = 0.0076241 kg/s

  

am i right ?