i) The normal force between the sphere and the particle just after the impulse is

ii) Minimum value of u for which the particle does not slip on the sphere is

iii) The angle made by the radius of the particle is

Given

The radius of the sphere on which the particle is kept is R and the horizontal speed is taken as u.

Formula Used

Using the conservation of static and dynamic energy such as centripetal and potential energy, we have the conservation equation as

where

m is the mass of the object, v is the velocity, R is the radius of the circular path, r is the radius of the object, h is the height, N is the reactionary force of the block.

Explanation

a) When a particle is kept on the top of the sphere a downward force of “mg” and an upward force of “N” is applied on the block giving the equation of forces as

Therefore, the normal force of the particle is taken as

b) When a particle is at minimum velocity, the reactionary force becomes zero

c) let us take the velocity as half as written in the question

Hence the new velocity is

Now calculating the angle

The velocity we use is u’ making the equation as

u’ is the velocity of the particle leaving the sphere

Placing the values of velocities as and