Question

A particle of mass $12$ kg is acted upon by a force $f$ $=\left(100-2x^2\right)$ when $f$ is in newton and $x$ is in metre. Calculate the wrok done by this force in moving the particle $x=0$ to $x=-10m$. What will be the speed at $x=10m$ if it starts from rest.

Answer 1

Given Force$F=100-2x^2$

And we know work done $\Delta W=Fdx$ where dx=displacement.

Now integrating the above equation from limits $x=0$ to $x=-10$,we get,

$W={\int }_0^{-10}(100-2x^2)dx=|{100x|}_0^{-10}-{|\frac{2}{3}(x^3|}_0^{-10}=\frac{-3000+2000}{3}=-333.33J$ negative sign implies work is done by the system.

Also we know $W=Fs$ where s=displacement=$vt$ where v=velocity and t= time.

And Force $F=ma=m\frac{v}{t}$ given initial velocity u=0 m/s.

So work done $W=m{v}^2$

$\sqrt{\frac{333.33}{12}}=v$

$v=5.27m/s$