Frequency of Oscillation of a Mass-Spring System – Rankers Physics

Dimensions: Practice Problem & Solution

The frequency of vibration (f) of a mass (m) suspended from a spring of spring constant (k) is given by a relation \(f = a.m^x k^y\), where (a) is a dimensionless constant. The values of (x) and (y) are: [1990]
\(x = \frac{1}{2}, y = \frac{1}{2}\)
\(x = -\frac{1}{2}, y = \frac{1}{2}\)
\(x = \frac{1}{2}, y = -\frac{1}{2}\)
\(x = -\frac{1}{2}, y = -\frac{1}{2}\)

Solution Explained:

To solve this problem, we apply the core principles of Dimensions. Understanding the underlying formula is key to arriving at the correct answer below:

Frequency \(f = [T^{-1}]\). Mass (m = [M]). Spring constant \(k = [MT^{-2}]\). Comparing dimensions of \(f = m^x k^y\): \([T^{-1}] = [M]^x [MT^{-2}]^y = [M^{x+y} T^{-2y}]\). Solving (x+y=0) and (-2y=-1) gives (y = 1/2) and (x = -1/2).

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