Question

Over a given region, a force (in newton) varies as
F=3x2-2x+1 In this region an object is displaced from
x =20cm to x,=40cm by the given force. Calculate
the amount of workdone.​

Answer 1

$\huge{\bold{\underline{\underline{....Answer....}}}}$

$\huge{\bold{\underline{Given:-}}}$

$\large{\bold{F = 3x^2 - 2x + 1}}$

$\huge{\bold{\underline{Explanation:-}}}$

As we know here the Force is changing w.r.t to "x"

Therefore,

we need to use the formula

$\large{\bold{W = \displaystyle\int F.s}}$

So, Substituting the values

$\large{W = \displaystyle\int 3x^2 - 2x + 1}$

Applying limits,

$\large{W = \displaystyle\int^{40}_{20} 3x^2 - 2x + 1}$

Integrating,

$\large{W = \left| \frac{3x^3}{3} - \frac{2x^2}{2} + x \right|^{40}_{20}}$

$\large{W = \left| \frac{ \cancel{3}x^3}{ \cancel{3}} - \frac{ \cancel{2}x^2}{ \cancel{2}} + x \right|^{40} _{20}}$

It becomes,

$\large{W = \left| x^3 - x^2 + x \right|^{40}_{20}}$

Substituting the values

$\large{W = (40)^3 - (40)^2 + 40 - [(20)^3 - (20)^2 + 20]}$

$\large{W = 64000 - 1600 + 40 - [8000 - 400 + 20]}$

$\large{W = 62440 - 7620}$

$\huge{\boxed{\boxed{W = 54820 \: J}}}$

So, the amount of work done is 54,820 J.