A ramp is constructed with a parabolic shape such that the height y at any point on its surface is - Physics - Laws Of Motion

Question

A ramp is constructed with a parabolic shape such that the height y
at any point on its surface is given in terms of its horizontal
distance x from the bottom of the ramp (x=y=0) by y=x^2/20 A small
block is to be set on the ramp The max height from the bottom level
at which the block can be kept on the ramp without sliding is
(Given that coefficient of static friction is 0 5)

Apurba Paul · Accepted Answer

Slope of a curve at any point is given by dydx So the slope of the
given curve is dydx=ddxx220=x10 So the angle of the slope is given
by tan-1x10
Now for block not to be slide the friction must be ≥ the
downward force Now in a slope the normal reaction is mgcosθ,
hence the frictional force is 0 5 mg cosθ And the
sliding force is mg sinθ now for condition for no slide
will be
0
5 mg cosθ≥mg sinθsinθ cosθ≥0
5tanθ≥0 5x10≥0
5x≥5so y=x220so maximum height is y=2520=1
25