Physical World Units and Measurements - Result Question 31
33. The frequency of vibration $f$ of a mass $m$ suspended from a spring of spring constant $k$ is given by a relation of the type $f=c m^{x} k^{y}$, where $c$ is a dimensionless constant. The values of $x$ and $y$ are
[1990]
(a) $x=\frac{1}{2}, y=\frac{1}{2}$
(b) $x=-\frac{1}{2}, y=-\frac{1}{2}$
(c) $x=\frac{1}{2}, y=-\frac{1}{2}$
(d) $x=-\frac{1}{2}, y=\frac{1}{2}$
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Answer:
Correct Answer: 33. (d)
Solution:
- (d) $f=c m^{x} k^{y}$;
Spring constant $k=$ force/length.
$[M^{0} L^{0} T^{-1}]=[M^{x}][MT^{-2}]^{y}=[M^{x+y} T^{-2 y}]$
$\Rightarrow x+y=0,-2 y=-1$ or $y=\frac{1}{2}$
Therefore, $x=-\frac{1}{2}$
The method of dimensions cannot be used to derive relations other than product of power functions.
