Question

derive by the method of dimensions, an expression for the energy E of a body,executing,S.H.M, assuming this energy depends upon the mass M, the frequency f, and amplitude A of the vibrating body.

Answer 1

let,
Energy = e
mass = m
frequency = f
Amplitude = a
then,
$e \propto \: {(m)}^{p} {(f)}^{q} {(a)}^{r} \\ so \\ (m {l}^{2} {t}^{ - 2} ) = {(m)}^{p}{(t)}^{ - q}{(l)}^{r} \\ since \: f \: = \frac{1}{t} \: and \: a \: have \: dimensions \: of \: length \\ so \\ p = 1 \\ q = 2 \\ r = 2 \\ so \: the \: relation \: is \\ e = k \: \cdot \: m \cdot \: {f}^{2} \cdot \: {a}^{2}$
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Answer 2

Answer:

Explanation:

Energy = e

mass = m

frequency = f

Amplitude = a

then,

e \propto \: {(m)}^{p} {(f)}^{q} {(a)}^{r} \\ so \\ (m {l}^{2} {t}^{ - 2} ) = {(m)}^{p}{(t)}^{ - q}{(l)}^{r} \\ since \: f \: = \frac{1}{t} \: and \: a \: have \: dimensions \: of \: length \\ so \\ p = 1 \\ q = 2 \\ r = 2 \\ so \: the \: relation \: is \\ e = k \: \cdot \: m \cdot \: {f}^{2} \cdot \: {a}^{2}