Question

A solid cylinder rolls up an inclined plane of inclination $\theta$ with an initial velocity $v$. How far does the cylinder go up the plane?

• A. $\frac{3v^{2}}{2g\, sin \,\theta}$
• B. $\frac{v^{2}}{4g\, sin \,\theta}$
• C. $\frac{3v^{2}}{g\, sin \,\theta}$
• D. $\frac{3v^{2}}{4g\, sin \,\theta}$

Answer 1

Answer: (D)

$\frac{3v^{2}}{4g\, sin \,\theta}$

Answer 2

Let the cylinder go up the plane upto a height $h$. Let $M$ and $R$ be the mass and radius of the cylinder respectively. According to law of conservation of mechanical energy, we get $\frac{1}{2}Mv^2 + \frac{1}{2} I\omega^2 = Mgh$ $\frac{1}{2}Mv^{2} +\frac{1}{2}\frac{MR^{2}}{2} \omega^{2} = Mgh$ $\left(\because {\text{For a solid cylinder}}, I= \frac{1}{2}MR^{2}\right)$ $\frac{1}{2}Mv^{2} +\frac{1}{4}MR^{2}\omega^{2} = Mgh$ $\frac{1}{2}Mv^{2}+\frac{1}{4}Mv^{2} = Mgh \quad\left(\because v= R\omega\right)$ $\frac{3}{4}Mv^{2} = Mgh$ $h= \frac{3v^{2}}{4 g}\quad...\left(i\right)$ Let $s$ be distance travelled by the cylinder up the plane.Then $sin \,\theta = \frac{h}{s}$ or $s = \frac{h}{sin\, \theta} = \frac{3\, v^{2}}{4\, g\, sin\, \theta}$ (Using$(i)$)