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Rotational Dynamics — Physics STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 SciencePhysicsRotational Dynamics4 Marks
Question
Starting from rest, an object rolls down along an incline that rises by $3$ units in every $5$ units (along it). The object gains a speed of $\sqrt{10} m / s$ as it travels a distance of $\frac{5}{3} m$ along the incline. What can be the possible shape/s of the object?
 
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Answer

Data $: \sin \theta=\frac{3}{5}, u =0, v =\sqrt{10} m / s , L =\frac{5}{3} m , g =10 m / s ^2$
$v=\sqrt{\frac{2 g L \sin \theta}{1+\left(k^2 / R^2\right)}}=\sqrt{\frac{2 g L \sin \theta}{1+\beta}}$
$\therefore v^2=\frac{2 g L \sin \theta}{1+\beta}$
$\therefore 1+\beta=\frac{2 g L \sin \theta}{v^2}=\frac{2(10)\left(\frac{5}{3}\right)\left(\frac{3}{5}\right)}{(\sqrt{10})^2}=2$
$\therefore \beta=\frac{k^2}{R^2}=1$
Therefore, the body rolling down is either a ring or a cylindrical shell.

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