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

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

Answer 1

Answer: (B)

For the smooth upper half, using $v^{2}-u^{2}=2as$ , we get
$v^{2}-0^{2}=2g\left(\right.sinϕ\left.\right)\times l$
$v=\sqrt{2 g \, \left(\right. sinϕ \left.\right) \times l}$
For the lower half, using $v^{2}-u^{2}=2as$ , we get
$0^{2}-2g \, sin \, \phi\times l=-2\left(\right. \left(\mu \right)_{k} \, cosϕ - sinϕ \left.\right)g\times l$
or $sinϕ=\mu _{k}cosϕ-sinϕ$
$\mu _{k}cosϕ=2sinϕ$
or $\mu _{k}=2tanϕ$