(A) The work done $W$ by a variable force $\vec F$ is given by the line integral: $W = \int \vec F \cdot d\vec r$.Given $\vec F = (2x\hat i + y\hat j)

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(A) The work done $W$ by a variable force $\vec F$ is given by the line integral: $W = \int \vec F \cdot d\vec r$.Given $\vec F = (2x\hat i + y\hat j)$ and $d\vec r = (dx\hat i + dy\hat j)$.$W = \int_{(0,0)}^{(1,2)} (2x\hat i + y\hat j) \cdot (dx\hat i + dy\hat j) = \int_{(0,0)}^{(1,2)} (2x\,dx + y\,dy)$.Integrating the terms separately:$W = \int_{0}^{1} 2x\,dx + \int_{0}^{2} y\,dy$.$W = [x^2]_{0}^{1} + [\frac{y^2}{2}]_{0}^{2}$.$W = (1^2 - 0^2) + (\frac{2^2}{2} - \frac{0^2}{2})$.$W = 1 + \frac{4}{2} = 1 + 2 = 3\,units$.

Class 11 Physics Work, Energy, Power and Collision Work Done by Variable Force and Force-Displacement Graph

Medium

$A$ body of mass $200\,g$ is moving along the $XY$ plane. The work performed by the force given by $\vec F = (2x\hat i + y\hat j)$ acting on it when the body is displaced from $(0, 0)$ to $(1, 2)$ will be equal to ............... $unit$.

A
$3$
B
$6$
C
$5$
D
$1.5$

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Work, Energy, Power and Collision

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Work Done by Variable Force and Force-Displacement Graph

If a force of $(6x^2 - 4x) \ N$ acts on a body of mass $10 \ kg$,then the work done by the force in displacing the body from $x = 2 \ m$ to $x = 4 \ m$ is: (in $J$)

MediumAP EAMCET 2025

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Power applied to a particle varies with time as $P = (3t^2 - 2t + 1) \text{ W}$. The change in kinetic energy of the particle from $t = 2 \text{ s}$ to $t = 4 \text{ s}$ is ............... $J$.

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$A$ force of $(5+3x) \text{ N}$ acts on a body of mass $20 \text{ kg}$ along the $x$-axis and displaces it from $x = 2 \text{ m}$ to $x = 6 \text{ m}$. The work done by the force is $.... \text{ J}$.

Medium

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$A$ body of $0.5 \,kg$ moves along the positive $X$-axis under the influence of a varying force $F$ (in newton) as shown below. If the speed of the object at $x=4 \,m$ is $3.16 \,ms^{-1}$,then its speed at $x=8 \,m$ is ................. $\,ms^{-1}$.

MediumKVPY 2011

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When a rubber band is stretched by a distance $x$,it exerts a restoring force of magnitude $F = ax + bx^2$,where $a$ and $b$ are constants. The work done in stretching the unstretched rubber band by $L$ is

MediumJEE MAIN 2014

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$A$ body of mass $200\,g$ is moving along the $XY$ plane. The work performed by the force given by $\vec F = (2x\hat i + y\hat j)$ acting on it when the body is displaced from $(0, 0)$ to $(1, 2)$ will be equal to ............... $unit$.

Similar Questions

If a force of $(6x^2 - 4x) \ N$ acts on a body of mass $10 \ kg$,then the work done by the force in displacing the body from $x = 2 \ m$ to $x = 4 \ m$ is: (in $J$)

Power applied to a particle varies with time as $P = (3t^2 - 2t + 1) \text{ W}$. The change in kinetic energy of the particle from $t = 2 \text{ s}$ to $t = 4 \text{ s}$ is ............... $J$.

$A$ force of $(5+3x) \text{ N}$ acts on a body of mass $20 \text{ kg}$ along the $x$-axis and displaces it from $x = 2 \text{ m}$ to $x = 6 \text{ m}$. The work done by the force is $.... \text{ J}$.

$A$ body of $0.5 \,kg$ moves along the positive $X$-axis under the influence of a varying force $F$ (in newton) as shown below. If the speed of the object at $x=4 \,m$ is $3.16 \,ms^{-1}$,then its speed at $x=8 \,m$ is ................. $\,ms^{-1}$.

When a rubber band is stretched by a distance $x$,it exerts a restoring force of magnitude $F = ax + bx^2$,where $a$ and $b$ are constants. The work done in stretching the unstretched rubber band by $L$ is

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