Question

A block is place on a ramp of parabolic shape given by the equation y = x²/20. If us = 0.5, what is the maximum height above the ground at which the block can be placed without slipping? (tan theta = us = dy/dx)​

Answer 1

Equation of the ramp is :

$\sf \: y = \dfrac{ {x}^{2} }{20}$

Coefficient of Static Friction = 0.5

Given Relation,

$\sf \: tan( \alpha ) = \dfrac{dy}{dx} = \mu$

Now,

$\sf \: \dfrac{dy}{dx} = \dfrac{2x}{20} \\ \\ \implies \sf \: \dfrac{dy}{dx} = \dfrac{x}{10} \\ \\ \implies \sf \: 0.5 = \dfrac{x}{10} \\ \\ \implies{\underline{ \boxed{\pink{ \sf \: x = {\frak{5 {\sf\: m}}}}}}{\:\star}}$

Thus,

$\longrightarrow \sf \: y = \dfrac{5 {}^{2} }{20} \\ \\ \longrightarrow {\underline{ \boxed{\purple{ \sf y = {\frak{1.25{\sf{ \: m}}}}}}}{\:\star}}$

The block should be placed at an height of 1.25 m from the ground so that it doesn't slip.

Answer 2

Answer:

1.25 m

Explanation:

Explanation on Attachment

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