Prove that the motion of a spring is SHM.

Asked by  | 23rd Mar, 2013, 01:33: PM

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  | 25th Mar, 2013, 10:52: AM

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