The maximum velocity and maximum acceleration of a particle performing a linear S.H.M. are α and β respectively. Then, path length of the particle is

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Maximum velocity of a particle in S.H.M. is given by $v_{ma x} = A ω = α$

Maximum acceleration is given by $a_{ma x} = A ω^{2} = β$

Squaring the maximum velocity, we get $A^{2} ω^{2} = α^{2}$

Dividing this by the maximum acceleration, we get $\frac{A ^{2} ω ^{2}}{A ω ^{2}} = \frac{α ^{2}}{β}$

$A = \frac{α ^{2}}{β}$

The path length of a particle in linear S.H.M. is $2 A$ .

Path length $= 2 A = \frac{2 α ^{2}}{β}$

Answer: $\frac{2 α ^{2}}{β}$

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