Question

The upper half of an inclined plane with inclination $\phi$ is perfectly smooth, while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom, if the coefficient of friction for the lower half is given by:

• A. $2\,sin\,\phi$
• B. $2\,cos\,\phi$
• C. $2\,tan\,\phi$
• D. $tan\,\phi$

Answer 1

Answer: (C)

$2\,tan\,\phi$

Answer 2

According to work-energy theorem, $W = \Delta K = 0$ ($\because$ Initial and final speeds are zero) $\therefore$ Work done by friction + work done by gravity $=0$ $-\mu\,mg\,cos\,\phi \frac{l}{2}+mgl\,sin\,\phi=0$ or $\frac{\mu}{2}\,cos\,\phi=sin\,\phi \therefore \mu=2\,tan\,\phi$