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Prove that the motion of a spring is SHM.

Asked by 23rd March 2013, 1:33 PM
Answered by Expert
Answer:
Consider a mass "m" attached to the end of an elastic spring. The other end of the spring is fixed
   at the a firm support as shown in figure "a". The whole system is placed on a smooth horizontal surface.
   If we displace the mass 'm' from its mean position 'O' to point "a" by applying an external force, it is    displaced by '+x' to its right, there will be elastic restring force on the mass equal to F in the left side    which is applied by the spring.
   According to "Hook's Law
F = - Kx ---- (1)
   Negative sign indicates that the elastic restoring force is opposite to the displacement.
   Where K= Spring Constant
   If we release mass 'm' at point 'a', it moves forward to ' O'. At point ' O' it will not stop but moves    forward towards point "b" due to inertia and covers the same displacement -x. At point 'b' once again    elastic restoring force 'F' acts upon it but now in the right side. In this way it continues its motion
   from a to b and then b to a.
   According to Newton's 2nd law of motion, force 'F' produces acceleration 'a' in the body which is given    by
F = ma ---- (2)
   Comparing equation (1) & (2)
ma = -kx
   Here k/m is constant term, therefore ,
a = - (Constant)x
or
a a -x
   This relation indicates that the acceleration of body attached to the end elastic spring is directly    proportional to its displacement. Therefore its motion is Simple Harmonic Motion.
Answered by Expert 25th March 2013, 10:52 AM
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