The upper half of an inclined plane with inclination φ 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

Question

The upper half of an inclined plane with inclination φ 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 φ (B)  2 tan φ (C) 2 cos φ (D) 2 sin φ

Tardigrade Admin · Accepted Answer

For the smooth upper half, using v2-u2=2as , we get v2-02=2g(.sinϕ.)× l v=√2 g (. sinϕ .) × l For the lower half, using v2-u2=2as , we get 02-2g sin φ× l=-2(. (μ )k cosϕ - sinϕ .)g× l or sinϕ=μ kcosϕ-sinϕ μ kcosϕ=2sinϕ or μ k=2tanϕ

Q. The upper half of an inclined plane with inclination $ϕ$ 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

$t an ϕ$

$2 t an ϕ$

$2 cos ϕ$

$2 s in ϕ$

Q. The upper half of an inclined plane with inclination ϕ 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

tanϕ

2tanϕ

2cosϕ

2sinϕ

For the smooth upper half, using v2−u2=2as , we get v2−02=2g(sinϕ)×l v=2g(sinϕ)×l​ For the lower half, using v2−u2=2as , we get 02−2gsinϕ×l=−2((μ)k​cosϕ−sinϕ)g×l or sinϕ=μk​cosϕ−sinϕ μk​cosϕ=2sinϕ or μk​=2tanϕ

Q. The upper half of an inclined plane with inclination $ϕ$ 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

$t an ϕ$

$2 t an ϕ$

$2 cos ϕ$

$2 s in ϕ$