A particular of mass m is executing oscillation about the origin on X- axis. Its potential energy is V(x) = K |x|³. Where K is a positive constant. If the amplitude of oscillation is a, then its time period T is proportional to

$\frac{1}{\sqrt{a}}$

a

$\sqrt{a}$

a^3/2

Solution

$\frac{1}{\sqrt{a}}$

Given, V = kx³

Total energy (E) = Maximum potential energy

= ka³

From conservation of energy

$\frac{1}{2} m {(\frac{dx}{dt})}^{2} + {kx}^{2} = {ka}^{3} \Rightarrow \frac{1}{2} m {(\frac{dx}{dt})}^{2} = k (a^{3} - x^{3}) \Rightarrow \frac{dx}{dt} = \sqrt{\frac{2 k}{m} (a^{3} - x^{3})} \frac{dx}{\sqrt{a^{3} - x^{3}}} = \sqrt{\frac{2 k}{m}} dt Or integrating \int_{0}^{a} \frac{dx}{\sqrt{a^{3} - x^{3}}} = \sqrt{\frac{2 k}{m}} \int_{0}^{T / 4} dt or \int_{0}^{a} \frac{dx}{\sqrt{a^{3} - x^{3}}} = \sqrt{\frac{2 k}{m}} \frac{T}{4} Let x = a \sin θ \Rightarrow dx = a \cos θ dθ ∴ \int_{0}^{π / 2} \frac{a \cos θ dθ}{\sqrt{a^{3} - (a \sin θ)^{3}}} = \sqrt{\frac{2 K}{m}} \frac{T}{4}$

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A particular of mass m is executing oscillation about the origin on X- axis. Its potential energy is V(x) = K |x|³. Where K is a positive constant. If the amplitude of oscillation is a, then its time period T is proportional to

$\frac{1}{\sqrt{a}}$

a

$\sqrt{a}$

a^3/2

Some More Questions From Oscillations Chapter

If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?

The displacement of an object attached to a spring and executing simple harmonic motion is given by x = 2 × 10^-2 cos πt metres. The time at which the maximum speed first occurs

A particle of mass m executes simple harmonic motion with amplitude ‘a’ and frequency ‘ν’. The average kinetic energy during its motion from the position of equilibrium to the end is

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency ω. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time

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A particular of mass m is executing oscillation about the origin on X- axis. Its potential energy is V(x) = K |x|3. Where K is a positive constant. If the amplitude of oscillation is a, then its time period T is proportional to 1a a a a3/2

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If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?

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A particle of mass m executes simple harmonic motion with amplitude ‘a’ and frequency ‘ν’. The average kinetic energy during its motion from the position of equilibrium to the end is

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency ω. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time

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Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

A particular of mass m is executing oscillation about the origin on X- axis. Its potential energy is V(x) = K |x|³. Where K is a positive constant. If the amplitude of oscillation is a, then its time period T is proportional to

$\frac{1}{\sqrt{a}}$

a

$\sqrt{a}$

a^3/2

The displacement of an object attached to a spring and executing simple harmonic motion is given by x = 2 × 10^-2 cos πt metres. The time at which the maximum speed first occurs

The function sin²(ωt) represents