A particle of mass m is allowed to oscillate near the point of minima of a vertical parabolic path having the equation x2=4ay , where x-axis is along the horizontal direction and y-axis is along the vertical direction. The angular frequency of small oscillations of the particle is

Question

A particle of mass m is allowed to oscillate near the point of minima of a vertical parabolic path having the equation x2=4ay , where x-axis is along the horizontal direction and y-axis is along the vertical direction. The angular frequency of small oscillations of the particle is (A) √(8 g/a) (B) √(2 g/a) (C) √(g/a) (D) √(g/2 a)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-kuzsaqahxkqi.png" /> textma = - textmg sin θ texta = - g sin θ or texta = - g tan θ ...(i) (as θ is small) Now, textx2 = 4 textay â´ ( textdy/ textdx) = ( textx/2 texta) â´ texta = - textg ( textx/2 texta) - ω 2 textx = - ( textgx/2 texta) ω = √( textg/2 texta)

$\frac{8 g}{a}$

$\frac{2 g}{a}$

$\frac{g}{a}$

$\frac{g}{2 a}$

$ma = - mg s in θ$
$a = - g s in θ$ or $a = - g t an θ$ ...(i)
(as $θ$ is small)
Now, $x^{2} = 4 ay$
$∴ \frac{dy}{dx} = \frac{x}{2 a}$
$∴ a = - g \frac{x}{2 a}$
$- ω^{2} x = - \frac{gx}{2 a}$
$ω = \frac{g}{2 a}$

Q. A particle of mass m is allowed to oscillate near the point of minima of a vertical parabolic path having the equation x2=4ay , where x−axis is along the horizontal direction and y−axis is along the vertical direction. The angular frequency of small oscillations of the particle is

a8g​​

a2g​​

ag​​

2ag​​

ma=−mgsinθ a=−gsinθ or a=−gtanθ ...(i) (as θ is small) Now, x2=4ay ∴dxdy​=2ax​ ∴a=−g2ax​ −ω2x=−2agx​ ω=2ag​​

$\frac{8 g}{a}$

$\frac{2 g}{a}$

$\frac{g}{a}$

$\frac{g}{2 a}$

$ma = - mg s in θ$
$a = - g s in θ$ or $a = - g t an θ$ ...(i)
(as $θ$ is small)
Now, $x^{2} = 4 ay$
$∴ \frac{dy}{dx} = \frac{x}{2 a}$
$∴ a = - g \frac{x}{2 a}$
$- ω^{2} x = - \frac{gx}{2 a}$
$ω = \frac{g}{2 a}$