CBSE Class 11-science Answered
Prove that the motion of a spring is SHM.
Asked by | 23 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. |
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According to "Hook's Law | |
F = - Kx ---- (1)
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Negative sign indicates that the elastic restoring force is opposite to the displacement. Where K= Spring Constant |
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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. |
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According to Newton's 2nd law of motion, force 'F' produces acceleration 'a' in the body which is given by | |
F = ma ---- (2)
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Comparing equation (1) & (2) | |
ma = -kx
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Here k/m is constant term, therefore , | |
a = - (Constant)x
or |
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a a -x
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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 | 25 Mar, 2013, 10:52: AM
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