Question

A wedge, as shown in the figure, has a rough curved surface with coefficient of surface with coefficient of kinetic friction $\frac{1}{\sqrt{3}}$ and whose shape can be given by equation $y=x^2$, Where the horizontal direction is $x$ - axis the vertical direction is $y$ - axis, the end of surface is $O$ as arigin and $x$ and $y$ are measured in metre. A small block is released from a certain point is released from a certain point on the surface. The maximum height of the point from where the block should be release so that it does not slip is $\frac{m}{n}$ meter. Find $mn$.
Diagram

Answer 1

Given,
Coefficient of kinetic friction, ${\mu }_k=\frac{1}{\sqrt{3}},andEquationofmotion,y=x^2$
As we know,
Slope of the graph is given by,
$\tan \theta =\frac{dy}{dx}$
For no slipping,
$\tan \theta =\mu \tan \theta =\frac{dy}{dx}=2x=\frac{1}{\sqrt{3}}\Rightarrow x=\frac{1}{2\sqrt{3}}\Rightarrow y=x^2=\frac{1}{12}m$

Final Answer: 12