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A particle P is initially at rest on the...

A particle `P` is initially at rest on the top `pf`a smooth hemispherical surface which is fixed on a horizontal plane. The particle is given a velocity `u` horizontally. Radius of spherical surface is `a`.
.

A

If the particle leaves the sphere, when it has fallen vertically by a distance of `(a)/(4)m u = (sqrt(ga))/(2)`.

B

If the particle leaves the sphere at angle `theta` (fig) where `cos theta = (sqrt(3))/(2)`, then `u = (sqrt(ag))/(3)`

C

If `u=0` and the particle just slides down the hemispherical surface, it will leave the surface when `cos theta = (2)/(3)`.

D

The minimum value of `u`, for the object to leave the sphere without sliding over the surface is `sqrt(ag)`.

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
Applying conservation of total energy
`(1)/(2) m u^(2) + mga (1 -c os theta) = (1)/(2) mv^(2)`
`mg cos theta -N =(mv^(2))/(a)`
for particle to lose contact `N = 0`
`v^(2) =ag cos theta , u^(2)+ga (2-3 cos theta) = 0`.
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