(C) The work done by a variable force is given by the integral $W = \int_{x_1}^{x_2} F \, dx$.Given $F = \alpha + \beta x^2$,the work done for a displ

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(C) The work done by a variable force is given by the integral $W = \int_{x_1}^{x_2} F \, dx$.Given $F = \alpha + \beta x^2$,the work done for a displacement from $x = 0$ to $x = 1 \ m$ is:$W = \int_{0}^{1} (\alpha + \beta x^2) \, dx = 5 \ J$.Integrating the expression:$W = [\alpha x + \frac{\beta x^3}{3}]_{0}^{1} = 5$.Substituting the limits:$(\alpha(1) + \frac{\beta(1)^3}{3}) - (0) = 5$.Given $\alpha = 1 \ N$,we have:$1 + \frac{\beta}{3} = 5$.$\frac{\beta}{3} = 4$.$\beta = 12 \ N/m^2$.

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

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$A$ force $F = \alpha + \beta x^2$ acts on an object in the $x$-direction. The work done by the force is $5 \ J$ when the object is displaced by $1 \ m$. If the constant $\alpha = 1 \ N$,then $\beta$ will be: (in $N/m^2$)

JEE MAIN 2025 All JEE MAIN

A
$15$
B
$10$
C
$12$
D
$8$

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Work, Energy and Power

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Work Done by Variable Force and Force-Displacement Graph

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$A$ toy car of mass $5 \, kg$ moves up a ramp under the influence of a force $F$ plotted against displacement $x$ as shown in the graph. The maximum height attained is given by:

Medium

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$A$ force $F = \alpha + \beta x^2$ acts on an object in the $x$-direction. The work done by the force is $5 \ J$ when the object is displaced by $1 \ m$. If the constant $\alpha = 1 \ N$,then $\beta$ will be: (in $N/m^2$)

Similar Questions

The value of the force constant for the given force-displacement graph is:

The adjacent figure shows the force-displacement graph of a moving body. What is the work done by this force in displacing the body from $x = 0$ to $x = 35\,m$ (in $,J$)?

$A$ particle moves along the $X$-axis from $x=0$ to $x=1 \, m$ under the influence of a force given by $F=3x^2+2x-10$. The work done in the process is ............. $J$.

$A$ force $\vec{F} = (7 - 2x + 3x^2) \, N$ is applied to an object. Calculate the work done as the object moves from $x = 0$ to $x = 5 \, m$. (in $, J$)

$A$ toy car of mass $5 \, kg$ moves up a ramp under the influence of a force $F$ plotted against displacement $x$ as shown in the graph. The maximum height attained is given by:

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