Question
Question: A body takes time \(t\) to reach the bottom of an inclined plane of angle θ will be horizontal. if t...
A body takes time t to reach the bottom of an inclined plane of angle θ will be horizontal. if the plane is rough, the time it takes now is 2t. The coefficient of friction of the rough surface is ?
Solution
In the question, a condition is already given and we can easily calculate the length of the incline and eventually after having two equations, we can find out the coefficient of friction of the rough surface.
Complete step by step answer:
As per the question, we can see the body takes time t to reach the bottom of an inclined plate of angle θ with the horizontal. Therefore,
Initial velocity, u=0
And acceleration down the incline, a=gsinθ
As the plane is initially smooth, the length of incline will be
s=21gsinθt2 ---(1)
Now when there is a presence of friction in the second case, let the coefficient of friction be μ
Therefore, the frictional force, f=μN=μmgcosθ
Here, m is the mass of the body
Now, by equating the force equation , we get acceleration as
a=(sinθ−μcosθ)g
Also, it is given that the time taken in the 2nd case is 2t
Therefore, the distance equation becomes
s=21×(sinθ−μcosθ)g×(2t)2 ---(2)
After solving equation one and two, we get
sinθ=(sinθ−μcosθ)4 ∴μ=43tanθ
Therefore the final answer to the solution is 43tanθ , so, the coefficient of friction in the second case is 43tanθ.
Note: If the initial situation was changed and there was no smooth surface then such problems are just put to trick students as the coefficient of friction of a body is constant and will be the same in all rough surfaces if we keep the rule of exception aside.