A wheel rotates for 5 seconds with a constant angular accelration and describes 100 radians during the time. If then rotates with a constant angular velocity and during next 5 seconds describe 80 radians. Find angular accelration of the wheel.

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Case} \mathrm{I}: (100 \mathrm{rad} \mathrm{in} \mathrm{first} 5 \mathrm{s}) \\ \mathrm{Angular} \mathrm{displacement}: \\ \mathrm{\theta }={\mathrm{\omega }}_0\mathrm{t}+\frac{1}{2}{\mathrm{\alpha t}}^2 \\ 100=5{\mathrm{\omega }}_0+\frac{1}{2}\mathrm{\alpha }{\left[5\right]}^2 \\ 100=5{\mathrm{\omega }}_0+12.5\mathrm{\alpha } ...... \left(1\right) \\ \mathrm{Case} \mathrm{II}: (80 \mathrm{rad} \mathrm{in} \mathrm{next} 5 \mathrm{s}) \\ \mathrm{Angular} \mathrm{displacement}: \\ \mathrm{\theta }=\mathrm{\omega t} \\ 80=5\mathrm{\omega } \\ \mathrm{\omega }=16 \mathrm{rad}/\mathrm{s} ...... \left(2\right) \\ \mathrm{For} \mathrm{Case} \mathrm{I}: \\ \mathrm{Angular} \mathrm{displacement},\mathrm{\theta }=\left[\frac{{\mathrm{\omega }}_0+\mathrm{\omega }}{2}\right]\mathrm{t} \\ 100=\left[\frac{{\mathrm{\omega }}_0+16}{2}\right]\left[5\right] \\ {\mathrm{\omega }}_0+16=40 \\ {\mathrm{\omega }}_0=24 \mathrm{rad}/\mathrm{s} \\ \mathrm{From} \mathrm{equation} \left(1\right), \\ \mathrm{\alpha }=\frac{100-5{\mathrm{\omega }}_0}{12.5} \\ =\frac{100-5\left[24\right]}{12.5} \\ =\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{ }\mathbf{rad}\mathbf{/}{\mathbf{s}}^{\mathbf{2}} \end{gathered}$$ ​

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