A particle of mass $m$ moves in a straight line. If $v$ is the velocity at a distance $x$ from a fixed point on the line and $v^{2} = a - b x^{2}$ , where $a$ and $b$ are constant then:

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Answer 1

Given,

$v^{2} = a - b x^{2}$
$∴ v^{2} + b x^{2} = a$
$\frac{x ^{2}}{a / b} + \frac{v ^{2}}{a} = 1$

For a S.H.M:
$x = A s i n (ω t + θ)$
$v = A ω c o s (ω t + θ)$
$\frac{x ^{2}}{A ^{2}} + \frac{v ^{2}}{A ^{2} ω ^{2}} = 1$
Comparing both equations:
$a / b = A^{2}$

$a = A^{2} ω^{2}$
$∴ b = ω^{2}$
$\Rightarrow ω = b$
$\Rightarrow f = \frac{ω}{2 π} = \frac{b}{2 π}$
And, the motion is simple harmonic

Answer 2

Given,

v^{2} = a - b x^{2}

∴ v^{2} + b x^{2} = a

\frac{x ^{2}}{a / b} + \frac{v ^{2}}{a} = 1

For a S.H.M:

x = A s i n (ω t + θ)

v = A ω c o s (ω t + θ)

\frac{x ^{2}}{A ^{2}} + \frac{v ^{2}}{A ^{2} ω ^{2}} = 1

Comparing both equations:

a / b = A^{2}

a = A^{2} ω^{2}

∴ b = ω^{2}

\Rightarrow ω = b

\Rightarrow f = \frac{ω}{2 π} = \frac{b}{2 π}

And, the motion is simple harmonic