Equation for velocity and energy at different positions of a vertical circular motion
Consider a body of mass 'm' performs vertical circular motion about the centre and radius 'r' as shown in figure As the motion is affected by the gravity the velocities of the body and tension in the string will be different at different points of the circle. Let vl be the velocity of the body at highest point. Let Tt be the tension at the highest points. The forces acting on the body at the highest position A are, i)
Weight of the body acting vertically downward direction, ii) Tension T in the string, acting vertically downward direction. Centripetal force acting on object at A is provided partly by weight and partly by tension in the string.
Similarly at lowest point B, the forces on body are, 1. Tension acting vertically upward direction T2 2. We ig h t mg ac t in g in v e r t ica lly d o w nw a r d direction
Where T 2 Is the tension and v 2 is velocity of object at B. I) Linear velocity of object at highest point A: There is certain velocity so called as critical velocity/minimum velocity (v 1 ) of object at A below which string become slack i.e. tension T1 vanishes..e (T=0)
Equation is required minimum velocity at highest point, so that string doesn't slack. If velocity at highest point is less than then string will be slackened and object will not be able to continue its circular motion. Object will perform vertical circular motion only if, its velocity at highest (top) point of circle is equal to or greater than.
II) Linear velocity at lowest point (B) : The decrease in potential energy between top -position A and bottom position B is,= mgr - (-mgr) = 2 mgr This must be equal to the increase in kinetic energy, when particle move from A
Equation gives required minimum velocity at lowest point. (bottom point B) so that it can safely travel along vertical circle of radius r. If velocity at lowest point is less than , then string will be slackened at some point and body will not be able to reach the top point to complete the circular motion.
III) Linear Velocity at Midway Point (C) : Total Energy at B = Total energy at C. (Law of conservation of energy) K.E. at B + P.E. at B K.E. at C + P.E. at C
This is an expression of velocity of particle at c Is the condition For proper circular motion Energies of particle in vertical circular motion Total energy of particle at highest point A = (Kinetic energy of particle at A)+ (Potential energy of particle at A) E = Ek+Ep
body possesses both K.E. and P.E. However
energy at every point is same i.e.
. Thus the total energy is conserved in vertical circular motion. Special Case : Difference in tensions at lowest and highest point.