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If 6 j of work is needed to stretch a spring from 9 cm to 11 cm and another 10 j is needed to stretch it from 11 cm to 13 cm, the natural length (in cm) of the spring will be 7cm
The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the system. The restoring force is often referred to in simple harmonic motion. The force which is responsible to restore original size and shape is called restoring force.
In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
6 = [tex]\int\limits^ {11-l}_ {9-l}\, kxdx[/tex]
6 = 1/2 k [tex]x^{2}[/tex] lm (9 - l) to (11 - l)
6 = 1/2 k [ [tex](11-l)^{2}[/tex] - [tex](9-l) ^{2}[/tex] ]
12/k = 40 - 4l equation 1
10 = [tex]\int\limits^ {13-l}_ {11-l}\, kxdx[/tex]
10 = 1/2 k [tex]x^{2}[/tex] lm (13 - l) to (11 - l)
20 k = 48 -4l equation2
solving equation 1 and 2 , we get
12/20 = 40 - 4 l / 48 -4 l
l = 7 cm
To learn more about hook's law here
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