1. NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential : due to the change in magnitude of velocity Tangential acceleration Radial acceleration NET acceleration * The NET acceleration is no longer pointing towards the centre of the circle. centre Speed is changing

2. Examples of non-uniform circular motions string  Vertical circle with a string and bob bob v Roller Coaster

3. string Vertical circle with a string and bob bob Free body diagram  mg cos  mg sin   mg T Radial direction : T - mg cos  = ma c = mv 2 / r Tangential direction : mg sin  = ma t Change in speed Change in direction

4. C an an object (mass m) go round a vertical circle of radius l if the initial speed at the bottom is u? u m l A B C D Can go round the circle : (1) Have enough energy to reach point C. (2) Have sufficient high centripetal force to maintain the circular motion at C. Consider Conservation of energy ;  0

5. C an an object (mass m) go round a vertical circle of radius l if the initial speed at the bottom is u? u m l A B C D Can go round the circle : (1) Have enough energy to reach point C. (2) Have sufficient high centripetal force to maintain the circular motion at C. T mg Consider force at point C ; v By Conservation of energy,  0

6. C an an object (mass m) go round a vertical circle of radius l if the initial speed at the bottom is u? u m l A B C D Can go round the circle : (1) Have enough energy to reach point C. (2) Have sufficient high centripetal force to maintain the circular motion at C. The object can go round the circle if the initial speed is greater than W hat happens if u < ?

7. u m l A B C D (1) < u < No more circular motion can be processed (as T = 0 but mg is greater than mv 2 /l) Can reach C (as u > ) Projectile motion due to gravity

8. u m l A B C D W hat happens if u < ? (2) < u < Between B and C(as u < ) Projectile motion due to gravity (3) u < Cannot reach B Swing about A between B and D For reaching B, 1/2 mu 2 = 1/2mv B 2 + mgl u 2  2gl u 

9. Consider the whole system (spaceship and man), Consider the man only, More about Circular Motion * A astronaut feels weightless in a spaceship which is moving with uniform circular motion about the Planet, say the Earth. man R mg R = 0 for weightless Mg + mg = (M+m) v 2 / r R mg v R Mg r v 2 = g r mg -R = mv 2 / r mg -R = m(g r) / r mg -R = mg R = 0

10. More about Circular Motion * Artificial gravity made for Space stations man Rotating axis   r R No weight as it is far away from all planets There is only normal contact reaction force due to contact N. R mg’ R = mg’

11. More about Circular Motion * Working principle of a centrifuge

12. P 1 = P P 2 = P+  P * Working principle of a centrifuge (1) Assume it is horizontally aligned with liquid of density  inside. (P 2 - P 1 )A Pressure gradient as centripetal force The pressure gradient increases with the distance from the rotating axis F C =  P A = (P 2 - P 1 )A = mr  2

13. * Working principle of a centrifuge an element of the liquid of density  (2) Consider an element of the liquid of density  inside. Net force = (P 2 - P 1 )A = [m] r  2 = [   V] r  2 =   r  r  2 Net force due to pressure gradient =  r A  2  r All liquid rotates with uniform speed

14. * Working principle of a centrifuge an element of other substance of density  ’ (2) Consider an element of other substance of density  ’ inside. ’’  ’<  for less dense object Move towards the axis F net Net force F net = (P 2 - P 1 )A =  r A  2  r F c Required centripetal force F c = [m’] r  2 = [  ’  V] r  2 =  ’   r  r  2 =  ’  r A  2  r  ’>  for denser object Move away from the axis

15. More about Circular Motion * Why centrifuge ? F C  =  ’  r A  2  r F net =  r A  2  r Excess force for separation  F c = (  ’  )  r A  2  r Assume  ’ >  Excess force for separation  F g = weight - upthrust Typical : r = 10 cm,  = 2500 rev min -1 ~ 700 / 1 = (  ’ A  r g  A  r g) = (  ’  ) A g  r