Exercise 2.4.5: Suppose we add possible friction to Exercise 2.4.4. Further, suppose you do not know the spring constant, but you have two reference weights 1 kg and 2 kg to calibrate your setup. You put each in motion on your spring and measure the frequency. For the 1 kg weight you measured 1.1 Hz, for the 2 kg weight you measured 0.8 Hz. a) Find k (spring constant) and c (damping constant). Find a formula for the mass in terms of the frequency in Hz. Note that there may be more than one possible mass for a given frequency. b) For an unknown object you measured 0.2 Hz, what is the mass of the object

a). C = b/2   and C = b/4

b).  

c). m = 63.4 kg (approx.)

Explanation:

Ex. 2.4.4

The total force acting on mass m is , where x is the displacement from the equilibrium position.

The equation of motion is

or     or   ,    where  

The solution is ,  where A and Ф are constants.

A is amplitude of  motion

is the angular frequency of motion

Ф is the phase angle.

Now,

or

Given

a).  

b).  

or  

Ex. 2.4.5

a). Total force acting on the mass m is

    The equation of motion is

    or ,  angular frequency of the undamped oscillation.

   γ = b/2m  is called the damping coefficient (γ=C)

   for 1 kg weight (= 9.8 N), = 1.1 Hz

    k = 4 x (3.14)² x (9.8) x 1.1²  = 4.6 x 10² N/m

  For 2 kg weight (= 19.6 N), = 0.8 Hz

    k = 4 x 9.8596 x 2 x 9.8 x 0.8²  = 5 x N/m

or

γ = b/2 (for 1 kg)  and   γ = b/4 (for 2 kg)

C = b/2   and C = b/4

b).

 For two particle problem,

where, μ is the reduced mass.

This time period is same for both the particles.

c).

         ( approx.)

it takes about 30 days for it to break down

less than one you can't create energy