Figure (8-E17) shows a smooth track which consists of a straight inclined part of length l joining smoothly with the circular part. A particle of mass m is projected up the incline from its bottom. (a) Find the minimum projection-speed u0 for which the particle reaches the top of the track. (b) Assuming that the projection-speed is 2u0 and that the block does not lose contact with the track before reaching its top, find the force acting on it when it reaches the top. (c) Assuming that the projection-speed is only slightly greater than u0, where will the block lose contact with the track?
i) The minimum projection-speed u0 for which the particle reaches the top of the track is 
ii) The projection-speed is 2u0 and that the block does not lose contact with the track before reaching its top, its force is 
iii) The block lose contact with the track at 
Given
The length of the inclined plane is “l”, mass of the particle is “m” and the velocity of the particle is “
”.
Formula Used
The formula for the total energy in terms of kinetic and potential energy is given as
where
The 
is the total energy in terms of kinetic and potential energy, m is the mass of the object, g is the acceleration in terms of gravity and l is the length of the object.
Explanation
a) The height of the particle is taken as
The potential energy
at the top of the sphere is
The total energy experienced on the particle is
The initial potential energy is 
The initial Kinetic energy is 
Therefore, the total energy of the particle is 
Hence, the equation of total energy is
b) The initial speed is 
Let the final speed be denoted as 
Hence, the total energy is equal to the sum of potential and kinetic energy.
The centripetal acceleration is given as
Hence, the force acted is
c) If the speed is doubled, we get the velocity as
. Hence, the angle made by the particle before leaving is
