A boy is initially seated on the top of a hemispherical ice mound of radius R 13

Question

A boy is initially seated on the top of a hemispherical ice

mound of radius R 13.8 m. He begins to slide down the ice, with

a negligible initial speed (Fig.). Approximate the ice as being

frictionless. At what height does the boy lose contact with the ice?

A boy is initially seated on the top of a hemispherical ice mound of radius R 13.8 m. He begins to slide down the ice, with a negligible initial speed (Fig.). Approximate the ice as being frictionless. At what height does the boy lose contact with the ice?

Explanation / Answer

Suppose boy looses the contact when he makes an angle (theta) with vertical axis, it's height is h and speed v

so,

Boy will loose contact if

mv^2/R = mg*cos(theta)

v^2/R = g*cos(theta)....(1)

By conservation of energy

1/2*mv^2 = mgR-mgh

V^2 = 2*g(R-h)

so, from (1)

2*g(R-h)/R = g*cos(theta)

==> 2(R-h)/R = cos(theta)

now, we can see by trogonometry that cos(theta) = h/R

so,

2(R-h)/R = h/R

==> 2(R-h) = h

h = 2R/3 = 2*13.8/3 = 9.2 m

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