A particle moves along the x-axis under the influence of a force F = -k * x + alpha * x^3. What is the potential energy function U(x) if U(0) = 0?

A. 12kx^2 - 14 x^4 — The potential energy is defined as U(x) = - integral of F dx from 0 to x. Integrating the given force function yields the expression.

JEE Main · PhysicsHard

$A particle moves along the x-axis under the influence of a force F = - k x + α x^{3} . What is the potential energy function U (x) if U (0) = 0 ?$

A. $\frac{1}{2} k x^{2} - \frac{1}{4} α x^{4}$
B. $- \frac{1}{2} k x^{2} + \frac{1}{4} α x^{4}$
C. $k x^{2} - α x^{4}$
D. $\frac{1}{2} k x^{2} + \frac{1}{4} α x^{4}$

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Correct answer: A — $\frac{1}{2} k x^{2} - \frac{1}{4} α x^{4}$

Solution

The potential energy is defined as U (x) = - \int_{0}^{x} F (x^{'}) d x^{'} . Integrating F = - k x + α x^{3} gives U (x) = \frac{1}{2} k x^{2} - \frac{1}{4} α x^{4} .

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A particle moves along the x-axis under the influence of a force F=−kx+αx3. What is the potential energy function U(x) if U(0)=0?

Solution

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More Work Energy Power practice questions

$A particle moves along the x-axis under the influence of a force F = - k x + α x^{3} . What is the potential energy function U (x) if U (0) = 0 ?$