Question

A sphere of radius$‘a’$ and mass $‘m’$ rolls along horizontal plane with constant speed$v_0$. It encounters an inclined plane at angle $\theta$ and climbs upward. Assuming that it rolls without slipping how far up the sphere will travel (along the incline)?

• A. $\frac{2}{5}\hspace{0.17em}\left(\frac{v_0^2}{g\sin \theta }\right)$
• B. $10\left(\frac{v_0^2}{7g\sin \theta }\right)$
• C. $\frac{v_0^2}{5g\sin \theta }$
• D. $7\left(\frac{v_0^2}{10g\sin \theta }\right)$

Answer 1

The correct option is D

$7\left(\frac{v_0^2}{10g\sin \theta }\right)$

Step 1: Given data and drawing the diagram

Radius$=a$

Mass $=m$

Speed$=v_0$

Acceleration due to gravity $=g$

Let height sphere climb $=h$

Let length of incline $=l$

Angular velocity $=\omega =\frac{v_0}{a}$

Step 2: Find how far up the sphere will travel

Potential energy is the energy of object due to its position and it is given by

Potential energy $=mgh$

Kinetic energy is the energy of an object due to its motion and it is given by

Kinetic energy $=\frac{1}{2}m{v}^2$

Friction produces heat, because the friction force carries out work when it acts. Heat due to friction is given by

Heat due to friction$=\frac{1}{2}I{\omega }^2$

From right triangle shown in figure,

$$\begin{gathered} \sin \theta =\frac{perpendicular}{hypotneus} \\ \sin \theta =\frac{h}{l} \\ \Rightarrow h=l\sin \theta \end{gathered}$$

By law of conservation of energy, the energy can neither be created nor be destroyed, it can transfer from one form to another. So,

Potential energy = kinetic energy+ heat due to friction

$\begin{array}{rcl}mgh& =& \frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2\\ mgh& =& \frac{1}{2}m{v}_0^2+\frac{1}{2}\times \frac{2}{5}m{a}^2\times \frac{v_0^2}{a^2}\\ gh& =& \frac{1}{2}{v}_0^2+\frac{1}{5}{v}_0^2\\ h& =& \frac{7}{10}\frac{v_0^2}{g}\end{array}$

Putting value of $h$, we get

$\begin{array}{rcl}l\sin \theta & =& \left(\frac{7}{10}\right)\left(\frac{v_0^2}{g}\right)\\ l& =& \left(\frac{7}{10}\right)\left(\frac{v_0^2}{g\sin \theta }\right)\end{array}$

Hence, option D is correct.