(D) The relationship between force $F$ and potential energy $U$ is given by $F = -\frac{dU}{dx}$.Therefore,$dU = -F dx$.Given $F = -2x$,we substitute

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(D) The relationship between force $F$ and potential energy $U$ is given by $F = -\frac{dU}{dx}$.Therefore,$dU = -F dx$.Given $F = -2x$,we substitute this into the equation:$dU = -(-2x) dx = 2x dx$.Integrating both sides with the condition that $U = 0$ at $x = 0$:$\int_0^U dU = \int_0^x 2x dx$.$U = [x^2]_0^x = x^2$.The potential energy function is $U = x^2$,which represents a parabola opening upwards with its vertex at the origin. This corresponds to the graph shown in option $D$.

Class 11 Physics Work, Energy, Power and Collision Potential Energy

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$A$ particle moves under the influence of a force $F = -2x$ in one dimension (where $x$ is the distance of the particle from the origin). Assume that the potential energy of the particle at the origin is zero. The schematic diagram of the potential energy $U$ as a function of $x$ is given by:

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Work, Energy, Power and Collision

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$A$ particle of mass $1 \ kg$ is free to move along the $x$-axis. Its potential energy is given by $U(x) = (\frac{x^2}{2} - x) \ J$. If the total mechanical energy of the particle is $2 \ J$,find the maximum speed of the particle.

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$A$ stick of mass $m$ and length $l$ is pivoted at one end and is displaced through an angle $\theta$. The increase in potential energy is ............

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$A$ particle moves under the influence of a force $F = -2x$ in one dimension (where $x$ is the distance of the particle from the origin). Assume that the potential energy of the particle at the origin is zero. The schematic diagram of the potential energy $U$ as a function of $x$ is given by:

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When an arrow is released from a bow,from where does the arrow acquire its kinetic energy?

$A$ uniform cable of mass $M$ and length $L$ is placed on a horizontal surface such that its $(1/n)^{th}$ part is hanging below the edge of the surface. To lift the hanging part of the cable up to the surface,the work done should be

$A$ particle of mass $1 \ kg$ is free to move along the $x$-axis. Its potential energy is given by $U(x) = (\frac{x^2}{2} - x) \ J$. If the total mechanical energy of the particle is $2 \ J$,find the maximum speed of the particle.

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