Question

A force varying with displacement$\left(x\right)$ is given as $F=a{e}^{-bx}$ acts on a particle of mass $m$ moving in a straight line. Find the work done on the particle in its displacement from origin to a distance $d$.

• A. $\frac{a}{b}(1-2e^{-bd})$
• B. $\frac{a}{b}(2-e^{-bd})$
• C. $\frac{a}{b}(1-e^{-bd})$
• D. $\frac{2a}{b}(2-e^{-bd})$

Answer 1

The correct option is C $\frac{a}{b}(1-e^{-bd})$

Work done by variable force,
$W=\underset{0}{\overset{d}{\int }}F.dx\dots 1$
here, $F=a{e}^{-bx}$
So,equation (1) becomes
$W=\underset{0}{\overset{d}{\int }}a{e}^{-bx}dx$
$=-\frac{a}{b}{\left[e^{-bx}\right]}_0^d$
$=\frac{a}{b}(1-e^{-bd})$