A) \[\frac{{{\alpha }^{2}}}{2\beta }\]
B) \[\frac{{{\alpha }^{2}}-{{\beta }^{2}}}{2\alpha }\]
C) \[\frac{{{\alpha }^{2}}-{{\beta }^{2}}}{2\beta }\]
D) \[\frac{\alpha (\alpha -\beta )}{2}\]
Correct Answer: A
Solution :
\[\omega =\frac{d\theta }{dt}=\alpha -\beta t\] or \[d\theta =\,(\theta -\beta t)\,dt\] when \[\omega =0,\,\,t=\frac{\alpha }{\beta }\] Now, inter grating Eq. (i) \[\int_{0}^{\theta }{d\theta =}\,\int_{0}^{t}{(\alpha -\beta t)\,dt}\] or \[\theta =\,\alpha \,[t]_{0}^{\alpha /\beta }-\beta \,\left[ \frac{{{t}^{2}}}{2} \right]_{0}^{\alpha /\beta }=\alpha \cdot \frac{\alpha }{\beta }-\beta \frac{{{\alpha }^{2}}}{2{{\beta }^{2}}}\] \[=\frac{{{\alpha }^{2}}}{\beta }-\frac{{{\alpha }^{2}}}{2\beta }=\frac{{{\alpha }^{2}}}{\beta }\]
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