Question

Hi! Please explain how to solve this step-by-step​

Answer 1

Question :

A force F = -k/x² (x ≠ 0) acts on a particle in x-direction. Find the work done by this force in displacing the particle from x = + a to x = + 2a. Here, k is a positive constant.

Solution :

❖ Work done is measured as the product of force and displacement.

But we can't use formula directly to solve this question as force is variable. We can directly use formula only when force is said to be constant.

Here we have to integrate the given force equation from a to 2a.

$\displaystyle\dag\:\underline{\boxed{\bf{\orange{W=\int\:F\cdot dx}}}}$

$\displaystyle\sf:\implies\:W=\int\:\left(-\dfrac{k}{x^2}\right)\cdot dx$

We know that, 1 / x² = x‾²

$\displaystyle\sf:\implies\:W=\int\:-kx^{-2}\cdot dx$

Rule : $\displaystyle\sf\int\:x^n\:dx=\dfrac{x^{n+1}}{n+1}$

$\sf:\implies\:W=\left[\dfrac{-kx^{-2+1}}{(-2+1)}\right]$

$\sf:\implies\:W=\left[\dfrac{-kx^{-1}}{(-1)}\right]$

$\sf:\implies\:W=[kx^{-1}]$

$\sf:\implies\:W=\left[\dfrac{k}{x}\right]_a^{2a}$

$\sf:\implies\:W=\dfrac{k}{2a}-\dfrac{k}{a}$

$\sf:\implies\:W=\dfrac{k-2k}{2a}$

$:\implies\:\underline{\boxed{\bf{\gray{W=-\dfrac{k}{2a}}}}}$

Answer 2

$\large{\bold{\underline{ Question -\: }}}$

A force F = - k/x² ( x ≠ 0) acts on a particle in x-direction . find the work done by his force in displacing the particle from x = + a to x = + 2a . here , k is a positive constant .

$\large{\bold{\underline{Answer - \: }}}$

$\sf \red{ :\longrightarrow\sf{\displaystyle \sf w= \frac{ - k}{ 2a}}}$

$\large{\bold{\underline{Step \: by \: step \: explanation - \: }}}$

⠀

$\sf{ dw = Fdx}$

⠀

$\sf :\longrightarrow{ \displaystyle\sf\int dw = \int Fdx}$

⠀

$\sf :\longrightarrow{ w= \displaystyle\sf\int_{x = a}^{x = 2a} Fdx}$

⠀

$\sf :\longrightarrow{ w= \displaystyle\sf\int_{ a}^{ 2a} \dfrac{ - k}{ {x}^{2} } dx}$

⠀

$\sf :\longrightarrow{ w= - k \displaystyle\sf\int_{ a}^{ 2a} {x}^{ - 2} dx}$

⠀

$\sf :\longrightarrow{\displaystyle w= - k\left[ \frac{ {x}^{ - 2} + 1}{ - 2 + 1} \right]_{a}^{a}}$

⠀

$\sf :\longrightarrow{\displaystyle\sf w= \frac{ - k}{ - 1} \left[ \frac{ 1}{ x} \right]_{2a}^{a}}$

⠀

$\sf :\longrightarrow{\displaystyle\sf w= k \: \frac{ 1}{ 2a} - \frac{1}{a}}$

⠀

$\sf \blue{ :\Longrightarrow{\displaystyle\sf w= \frac{ - k}{ 2a}}}$