Question

A ramp is constructed in parabolic shape such that the height $y$ of any point on its surface is given in terms of the point's horizontal distance $x$ from the bottom of the ramp by $y=x^2/2L$. A block of granite is to be set on the ramp the coefficient of static friction is $\mathrm{0.80.}$ What is the maximum $x$ coordinate $x_M$ at which the block can be placed on the ramp and remain at rest, if $L=10m?$

• A. 8 m
• B. 8.4 m
• C. 9 m
• D. 9.4 m

Answer 1

Answer: (A)

$y=\frac{x^2}{2L}\Rightarrow \frac{dy}{dx}=\left(\frac{x}{L}\right)=\tan \theta$
When the block is placed at a slope ' $\tan \theta$ ', the F.B.D. is as shown in the figure. At $x_M$
$Mg\sin \theta =f_{\max }=\mu N=\mu Mg\cos \theta$
$\Rightarrow \tan \theta =\mu$ or $\frac{x_M}{L}$
$\Rightarrow x_M=\mu \cdot L$
$\Rightarrow x_M=(0.8)(10m)=8m$