Question

Determine the value of the spring constant that would result in a spring-mass system that would execute one complete cycle of oscillation every 2 s, for a mass of 1 kg. What natural frequency does this system exhibit in radians/second?

Answer 1

Expt. No... Date_ 120 Solution: According to D'Alembert Rules mö + (Restoring force) co msi + 4% =0 x + k xao fan- t Spring-mass System Ko Spring Constant. m natural frequency, .. Time Period Given for spring- mass system - I = 25 ~ Time Period Nath wn= 3.14 rads Uncu As we know standard formula Wh = 2 Natural frequency I wm = 3.14 Pads | Now whak k= (3.14) ²x m Spring Constant (K = 9.8596 Now m 1kg KN Teacher's Signature .. Page No.

Answer 2

SOLUTION :

In a spring mass system , simple harmonic motion is experienced.

Let the movement of the mass, x = A sin ω t

=> dx/dt = A ω cos ω t 

=> d2x/dt2 = a = - A ω^2 sin ω t

Now the force of mass movement and spring force are equal and in opposite direction for equilibrium position.

=> m a = - k x 

=> m  (- A ω^2 sin ω t ) = - k (A sin ω t)

=> m ω^2 = k

=> ω = sqrt (k/m)  

=> T = 2 π / ω = 2 π / sqrt(k/m)  = 2 

=> k/m = π^2

=> k = spring constant = m π^2 = 1 * 3.14^2  kg =  9.86 N/m (ANSWER)

Now,

Natural frequency 

= sqrt (k/m) 

= sqrt ( π^2 )

= 3.14 rad/sec (ANSWER).