Q62 of 101 Page 132

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








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