non-uniform circular motion * there are two components of acceleration: radial / centripetal : due...
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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
Examples of non-uniform circular motions
string
Vertical circle with a string and bob
bob
v
Roller Coaster
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
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
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
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
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
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
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
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
More about Circular Motion
* Working principle of a centrifuge
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
* 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
* 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
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