(B) The force $F$ acting on the particle is given by the negative gradient of the potential energy: $F = -\frac{dU}{dr}$.Given $U = \alpha r^{-4} - \b

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(B) The force $F$ acting on the particle is given by the negative gradient of the potential energy: $F = -\frac{dU}{dr}$.Given $U = \alpha r^{-4} - \beta r^{-5}$.Differentiating with respect to $r$:$F = -\frac{d}{dr}(\alpha r^{-4} - \beta r^{-5}) = -[\alpha(-4r^{-5}) - \beta(-5r^{-6})] = 4\alpha r^{-5} - 5\beta r^{-6}$.For the particle to be in equilibrium,the net force must be zero: $F = 0$.$4\alpha r^{-5} - 5\beta r^{-6} = 0$.$4\alpha r^{-5} = 5\beta r^{-6}$.Multiplying both sides by $r^6$:$4\alpha r = 5\beta$.Therefore,$r = \frac{5\beta}{4\alpha}$.

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

Medium

Potential energy of a particle varies with position $r$ as,$U = \left( \frac{\alpha}{r^4} - \frac{\beta}{r^5} \right) \text{ J}$,where $\alpha$ and $\beta$ are positive constants. The particle will be in equilibrium at $r = \dots$

A
$\frac{4\alpha}{5\beta}$
B
$\frac{5\beta}{4\alpha}$
C
$-\frac{4\alpha}{5\beta}$
D
$-\frac{5\beta}{4\alpha}$

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Class 11

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Work, Energy, Power and Collision

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Potential Energy

$A$ uniform chain of mass $M$ and length $L$ is lying on a smooth horizontal table,with half of its length hanging down. The work done in pulling the entire chain up onto the table is

DifficultAP EAMCET 2020

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$A$ carpet of mass $M$ made of a material is rolled along its length in the form of a cylinder of radius $R$ and kept on a rough floor. If the carpet is unrolled without sliding to a radius $R/2$,the change in potential energy is ($g=$ acceleration due to gravity).

MediumMHT CET 2024

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$A$ particle with constant total energy $E$ moves in one dimension in a region where the potential energy is $U(x)$. The speed of the particle is zero where

Medium

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$A$ particle which is constrained to move along the $x-$axis is subjected to a force in the same direction which varies with the distance $x$ of the particle from the origin as $F(x) = -kx + ax^3$. Here $k$ and $a$ are positive constants. For $x \ge 0$,the functional form of the potential energy $U(x)$ of the particle is

DifficultIIT 2002

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$A$ uniform chain of length $L$ and mass $M$ is lying on a smooth table and one-third of its length is hanging vertically down over the edge of the table. If $g$ is the acceleration due to gravity, the work required to pull the hanging part onto the table is

DifficultIIT 1985

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Potential energy of a particle varies with position $r$ as,$U = \left( \frac{\alpha}{r^4} - \frac{\beta}{r^5} \right) \text{ J}$,where $\alpha$ and $\beta$ are positive constants. The particle will be in equilibrium at $r = \dots$

Similar Questions

$A$ uniform chain of mass $M$ and length $L$ is lying on a smooth horizontal table,with half of its length hanging down. The work done in pulling the entire chain up onto the table is

$A$ carpet of mass $M$ made of a material is rolled along its length in the form of a cylinder of radius $R$ and kept on a rough floor. If the carpet is unrolled without sliding to a radius $R/2$,the change in potential energy is ($g=$ acceleration due to gravity).

$A$ particle with constant total energy $E$ moves in one dimension in a region where the potential energy is $U(x)$. The speed of the particle is zero where

$A$ uniform chain of length $L$ and mass $M$ is lying on a smooth table and one-third of its length is hanging vertically down over the edge of the table. If $g$ is the acceleration due to gravity, the work required to pull the hanging part onto the table is

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