InterviewSolution
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InterviewSolution
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A body of mass `m` hung at one end of the spring executes simple harmonic motion . The force constant of a spring is `k` while its period of vibration is `T`. Prove by dimensional method that the equation `T = 2 pi m // k` is correct. Dervive the correct equation , assuming that they are related by a power law. |
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Answer» Correct Answer - ` C sqrt((m)/(k))` The given equation is `T = ( 2 pi m)/( k)` Taking the dimensions of both sides, we have `[T] = ([M])/( [ ML^(0)T^(-2)] = T^(2)` As the dimensions of two sides are not equal , hence the equation is incorrect. Let the correct relation be `T = Cm^(a) k^(b) , where C` is constant. Equating the dimensions of both sides , we get `[T] = [M]^(a) [MT^(-2)]^(b)` or `[M^(0) L^(0) T] = [M^(a+b)L^(0)T^(-2b)]` Comparing the powers of M,L, and T on both sides , we get ` a + b = 0 and -2b = 1`. Therefore , `b = -(1)/( 2) and a = (1)/(2)` :. `T = C m^(-1//2) k ^(-1//2) = C sqrt((m)/(k))` This is the correct equation. |
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