Answer :

1732.5 (Range : 675 to 700 or 1725 to 1740)

The equation of speed with respect to time is given by

N = 125 t + 500 for 4 sec ≤ t ≤ 8 sec

We know that,

$\mathrm{J}\frac{\mathrm{d}\omega }{\mathrm{d}\mathrm{t}}={\mathrm{T}}_{\mathrm{e}}-{\mathrm{T}}_{\mathrm{L}}$

⇒${\mathrm{T}}_{\mathrm{e}}=\mathrm{J}\frac{\mathrm{d}\omega }{\mathrm{d}\mathrm{t}}+{\mathrm{T}}_{\mathrm{L}}$

⇒$\omega {\mathrm{T}}_{\mathrm{e}}=\mathrm{J}\omega \frac{\mathrm{d}\omega }{\mathrm{d}\mathrm{t}}+\omega {\mathrm{T}}_{\mathrm{L}}$

⇒${\mathrm{P}}_{\mathrm{e}}=\mathrm{J}\omega \frac{\mathrm{d}\omega }{\mathrm{d}\mathrm{t}}+\omega {\mathrm{T}}_{\mathrm{L}}$

⇒$\frac{\mathrm{d}\mathrm{E}}{\mathrm{d}\mathrm{t}}=\mathrm{J}\omega \frac{\mathrm{d}\omega }{\mathrm{d}\mathrm{t}}+\omega {\mathrm{T}}_{\mathrm{L}}$

⇒ dE = jωdω + TLωdt

⇒ E = J∫ωdω + TL∫ωdt

⇒$\mathrm{E}=\mathrm{J}{\left(\frac{2\pi }{60}\right)}^2{\int }_{1000}^{1500}\mathrm{N}\mathrm{d}\mathrm{N}+{\mathrm{T}}_{\mathrm{L}}\frac{2\pi }{60}{\int }_4^8\mathrm{N}\mathrm{d}\mathrm{t}$

$=\mathrm{J}{\left(\frac{2\pi }{60}\right)}^2{\left(\frac{{\mathrm{N}}^2}{2}\right)}_{1000}^{1500}+2.\frac{2\pi }{60}{\int }_4^8(125\mathrm{t}+500)\mathrm{d}\mathrm{t}$

$=0.1\times {\left(\frac{2\pi }{60}\right)}^2\left(\frac{{1500}^2-{1000}^2}{2}\right)+2.\frac{2\pi }{60}{[\frac{125t^2}{2}+500t]}_4^8$

=685.339 + 1047.197

= 1732.5865 J