In figure, a crate slides down an inclined right angled through. The coefficient of kinetic friction between the crate and the trough is \[{{\mu }_{k}}.\] The acceleration of the crate is: |
A) \[\operatorname{g}\,sin\theta \]
B) \[\operatorname{g}\,sin\theta -{{\mu }_{k}}g\cos \theta \]
C) \[\operatorname{g}\,(sin\theta -\sqrt{2}{{\mu }_{k}}\cos \theta )\]
D) None of these
Correct Answer: C
Solution :
If \[N\]is the normal reaction of each block, then | |
\[2N\cos 45{}^\circ =mg\cos \theta \] | |
\[\therefore \]\[N=\frac{mg\cos \theta }{\sqrt{2}}\] | |
Frictional force on the block | |
\[f-{{\mu }_{\operatorname{k}}}\left( 2N \right)\] | |
\[=\sqrt{2}{{\mu }_{k}}\operatorname{mg}\cos \theta \] | ...(i) |
Now \[mg\sin \theta -f=ma\] | ...(ii) |
From above equations, we get | |
\[a=g\left( \sin \theta -\sqrt{2}{{\mu }_{k}}\cos \theta \right).\] |
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