NONuniform Circular Motion The NET acceleration is no

NON-uniform Circular Motion * The NET acceleration is no longer pointing towards the centre of the circle. * 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 Radial acceleration centre Tangential acceleration NET acceleration Speed is changing

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

Free body diagram Change in direction Vertical circle with a string and bob Radial direction : T - mg cos q = mac = mv 2 / r string q T bob Tangential direction : mg sin q = mat mg sin q q mg mg cos q Change in speed

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

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

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

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

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

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

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

More about Circular Motion * Working principle of a centrifuge

* Working principle of a centrifuge (1) Assume it is horizontally aligned with liquid of density P 1 = P r inside. P 2 = P+ P (P 2 - P 1)A Pressure gradient as centripetal force FC = P A = (P 2 - P 1 )A = mrw 2 The pressure gradient increases with the distance from the rotating axis

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

* Working principle of a centrifuge (2) Consider an element of other substance of density r’ inside. r’ r’< r for less dense object Net force Fnet = (P 2 - P 1 )A = r r A w 2 r Required centripetal force Fc = [m’] r w 2 = [r’ V] r w 2 = r’(A r) r w 2 = r’ r A w 2 r Move towards the axis r’> r for denser object Move away from the axis

More about Circular Motion * Why centrifuge ? Excess force for separation = weight - upthrust = (r’ A r g ) - (r A r g) FC = r’ r A w 2 r Fnet = r r A w 2 Fg = (r’ - r) A g r r Assume r’ > r Excess force for separation = (r’ - r) r A w 2 r Fc Typical : r = 10 cm, w = 2500 rev min-1 ~ 700 / 1