Unstretched spring length?

Question

When a 340g spring is stretched to a total length of 40cm, it supports transverse wave propagating at 4.5m/s. When it's stretched to 60cm, the waves propagate at 12m/s.

a) Find the spring constant
b) what is the original length of the unstretch spring length?

Full points for clear workings and explanations

Answer 1

μ = m/L

Here, m is the mass of the spring and L is the length of the original length of the unstretch spring length.

Substitute 0.34 kg for m .

μ = 0.34/L

Write the equation of the force in spring.

F1 = k(x1 - L)

Write the expression for speed .

v1 = √ (F1/μ)

v1 = √ (k(x1 - L)/m/L)

v1^2=kL(x1 - L)/m                                      ..... (a)

--------------------------

When propagate speed (V2)

F2 = k(x2 - L)

Write the expression for speed .

v2 = √ (F2/μ)

v2 = √ (k(x2 - L)/m/L)

v2^2=kL(x2 - L)/m                                      ..... (b)

-------------------------

b)

Divide equation (b) by equation (a)

v2^2/v1^2=(kL(x2 - L)/m)   / (kL(x1 - L)/m)

v2^2/v1^2=(x2 - L) / (x1 - L)

Substitute 4.5 m/s for v1,12 m/s for v2, 0.4 m for x1, and 0.6 m for x2.

12^2/4.5^2=(0.6 - L) / (0.4 - L)

7.11=(0.6 - L) / (0.4 - L)

2.844-7.11L=0.6-L

L= 0.367 m

   =36.7 cm

--------------------------------

a)

From the equation (b)

v2^2=kL(x2 - L)/m

12^2=k(0.367)(0.6 -0.367)/0.34

k=572 N/m