Question

The 20 g head of a bobble-head doll oscillates in SHM at a frequency of 3.6 Hz .

A) What is the spring constant of the spring on which the head is mounted?

Express your answer to two significant figures and include the appropriate units.

B) 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?

Express your answer using two significant figures.

Answer 1

(a) The frequency of oscillation of the spring is given by ,

  

2.V m

where, m is the mass of the head = 20 g = 20/1000 = 0.02 kg.

K is the spring constant.

F is the frequency = 3.6 Hz.   

Therefore,

k= 47?f'm

k = 47° *3.6*.02

k= 10.23N/m
...........(Ans.)

(b) The amplitude of heads oscillation is expressed as a function of time as follows

  

A(t) = A.e-bt/2m

A(t) - Amplitude of oscillation at time 't'

A₀ - initial amplitude of oscillation

b is the damping constant.

The amplitude reduces due to the drag force.

Taking logarithm,

In (Alt)/A) = -bt/2m

b=(-2m/t) * In (Alt)/40)

...........(Ans.)

b = 0.01386kgs