A ball is released from rest from point P of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is α with respect to point P. Which of the following graphs represent the correct relation between A and α when ball goes from Q to R?
The Correct Option is C
Solution and Explanation
\(N=mg\ sin\ α+ \frac {mv^2}{r}\)
and, \(v^2 = 2g × R\ sin\ α\)
\(∴N = mg\ sin\ α + m × (2g\ sin\ α)\)
\(= 3mg\ sin\ α\)
∴ratio, \(A=\frac {\frac {mv^2}{R}}{N}\)
\(=\frac {2mg\ sin\ α}{3mg\ sin\ α}\)
\(=\frac {2}{3}\)
So,the corrct option is \((C)\):
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Concepts Used:
Laws of Motion
The laws of motion, which are the keystone of classical mechanics, are three statements that defined the relationships between the forces acting on a body and its motion. They were first disclosed by English physicist and mathematician Isaac Newton.
Newton’s First Law of Motion
Newton’s 1st law states that a body at rest or uniform motion will continue to be at rest or uniform motion until and unless a net external force acts on it.
Newton’s Second Law of Motion
Newton's 2nd law of motion deals with the relation between force and acceleration. According to the second law of motion, the acceleration of an object as built by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
Newton’s Third Law of Motion
Newton's 3rd law of motion states when a body applies a force on another body that there is an equal and opposite reaction for every action.