non-uniform circular motion * there are two components of acceleration: radial / centripetal : due...

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NON-uniform Circular Motion re TWO components of acceleration: l / centripetal : due to the change in direction of ntial : due to the change in magnitude of velocity Tangential acceleration Radial acceleration NET acceleration NET acceleration is no longer pointing towards e centre of the circle. centre Speed is changing

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Page 1: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

NON-uniform Circular Motion

* There are TWO components of acceleration:Radial / centripetal : due to the change in direction of velocityTangential : due to the change in magnitude of velocity

Tangential accelerationRadial acceleration

NET acceleration

* The NET acceleration is no longer pointing towards the centre of the circle.

centre

Speed is changing

Page 2: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

Examples of non-uniform circular motions

string

Vertical circle with a string and bob

bob

v

Roller Coaster

Page 3: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

string

Vertical circle with a string and bob

bob

Free body diagram

mg cos

mg sin

mg

T

Radial direction : T - mg cos = mac = mv2 / r

Tangential direction : mg sin = mat

Change in speed

Change in direction

Page 4: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

Can an object (mass m) go round a vertical circle of radius l if the initial speed at the bottom is u?

um

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 ;

)2(2

1

2

1 22 lmgmvmu

glvu 422 gluv 422

glu 42 glu 4

)2(222 lgvu

0

Page 5: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

Can an object (mass m) go round a vertical circle of radius l if the initial speed at the bottom is u?

um

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.

l

mvTmg

2

glglu 42

glu 5

T

mg

Consider force at point C ;

v

mgl

mvT

2

glv 2

By Conservation of energy, gluv 422

0

Page 6: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

Can an object (mass m) go round a vertical circle of radius l if the initial speed at the bottom is u?

um

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.

glu 4

glu 5

The object can go round the circle if the initial speed is greater than

gl5

What happens if u < ?gl5

Page 7: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

um

l

A

B

C

D

What happens if u < ?gl5

(1) < u < gl4 gl5

No more circular motion can be processed (as T = 0 but mg isgreater than mv2/l)

Can reach C (as u > )gl4

Projectile motion due to gravity

Page 8: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

um

l

A

B

C

D

What happens if u < ?gl5

(2) < u < gl2 gl4

gl4 Between B and C(as u < )

Projectile motion due to gravity

(3) u < gl2

Cannot reach B

Swing about A between B and DFor reaching B,1/2 mu2 = 1/2mvB

2 + mgl u2 2gl u gl2

Page 9: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

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.

manR

mg

R = 0 for weightless

Mg + mg = (M+m) v2 / r

R

mg

v

R

Mg

r

v2 = g r

mg -R = mv2 / r

mg -R = m(g r) / r

mg -R = mg

R = 0

Page 10: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

More about Circular Motion

* Artificial gravity made for Space stations

manRotating axis

r

R

No weight as it is far away from all planetsThere is only normal contact reaction force due to contact N.

2mrR

R

mg’

R = mg’

'mg22 8.9' msgr

Page 11: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

More about Circular Motion

* Working principle of a centrifuge

Page 12: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

P1 = P P2 = P+P

* Working principle of a centrifuge

(1) Assume it is horizontally aligned with liquid of density inside.

(P2 - P1)A

Pressure gradient as centripetal forceThe pressure gradient increases

with the distance from the rotating axis

FC = P A = (P2 - P1 )A = mr2

Page 13: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

* Working principle of a centrifuge

(2) Consider an element of the liquid ofan element of the liquid of density density inside.

Net force = (P2 - P1 )A

= [m] r 2

= [ V] r 2

= rr 2

Net force due to pressure gradient = r A 2 r

All liquid rotates with uniform speed

Page 14: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

* Working principle of a centrifuge

(2) Consider an element of other substance ofan element of other substance of density density ’’ inside.

’< for less dense object

Move towards the axis

Net force FFnetnet = (P2 - P1 )A = r A 2 r

Required centripetal force FFcc = [m’] r 2

= [’ V] r 2

= ’ rr 2 = ’r A 2 r’> for denser object

Move away from the axis

Page 15: NON-uniform Circular Motion * There are TWO components of acceleration: Radial / centripetal : due to the change in direction of velocity Tangential :

More about Circular Motion

* Why centrifuge ?

FC=’r A 2 r

Fnet =r A 2 r

Excess force for separation Fc

= (’)r A 2 r

Assume ’ >

Excess force for separation Fg

= weight - upthrust

rAg

rAr

F

F

g

c

'

' 2

g

r 2

Typical : r = 10 cm, = 2500 rev min-1

~ 700 / 1g

c

F

F

= (’ A r gA r g)

= (’) A g r