Question

In a children’s park, there was a slide to be made by a contract. By mistake, the person who had taken the contract made the coefficient of friction of the slide as high as $\dfrac{1}{3}$. Now, the fun is that the child expecting to slide down the incline will stop somewhere in between. Find the angle $\theta $ with the horizontal at which he will stop he will stop on the incline. (Assume negligible frictional force)

Answer 1

Hint: To solve this question we need to understand that we have to find that value of $\theta $ in which the child will not slide down. So in such a case we have to apply the formula and do some simplification and we get the required answer.

Formula used:
$ma = mg\sin \theta - \mu mg\cos \theta = 0$

Complete step by step answer:
It is given in the question that the coefficient of friction is $\dfrac{1}{3}$, therefore we can write the value of $ \Rightarrow \mu = \dfrac{1}{3}$.
Incorporating the values in the above formula we get-
$ \Rightarrow mg\sin \theta - \dfrac{1}{3}mg\cos \theta = 0$
Moving $\dfrac{1}{3}mg\cos \theta $ on the right hand side we get-
$ \Rightarrow mg\sin \theta = \dfrac{1}{3}mg\cos \theta $
Now $mg$ gets cancelled from both the sides and we get-
$ \Rightarrow \sin \theta = \dfrac{1}{3}\cos \theta $
Therefore we can write-
$ \Rightarrow \tan \theta = \dfrac{1}{3}$
$ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{1}{3}} \right)$
On using the calculator we get the value of ${\tan ^{ - 1}}\left( {\dfrac{1}{3}} \right)$
$ \Rightarrow {18.43^ \circ }$

Thus the required value of $\theta = {18.43^ \circ }$.

Note: For solving this sum we need to remember the formulas of trigonometry also. Most of the students make mistakes in calculation while solving this question. You can also solve this question by applying the concept of free body diagram.
It is important to note that coefficient of friction is a scalar quantity. It does not have any unit.
Moreover, it does not have any dimension. It totally depends upon the objects that are causing frictions.