Q. 64

A smooth sphere of radius R is made to translate in a straight line with a constant acceleration α. A particle kept on the top of the sphere is released from there at zero velocity with respect to the sphere. Find the speed of the particle with respect to the sphere as a function of the angle θ it slides.

Answer :

The speed of the particle with respect to sphere is


Given


The radius of the smooth sphere is R, the constant acceleration 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, is the angle of exit.


Explanation


As the sphere is moving at a constant acceleration with a pseudo force acting “” on the particle opposite to the downward force is .


Therefore, the speed of the particle is


The tangential force =



Placing the acceleration as




Integrating both sides we get



To know C, we apply boundary pressure of and for which we get:




Hence, the expression of velocity is




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