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17 Motion in a circle
AQA Physics A level pp 274-285
Uniform circular motion
• Speed unchanged
• Speed = |Velocity| = v
2r
v
T
– Where r = radius and T = period
• Velocity changes since direction is changing
2 r
v
T
• Frequency = 1/T
 v  2rf
Angular velocity, ω
•
•
•
•
Rotational frequency f = 1/T
Angular frequency = 360o per second
We like to have simple units
We like equations without (too many)
constants v  2rf
• Define
  2f
• So
v  r
• Also

d


t

dt
Radians
•
•
•
•
•
Units of ω are radians per second
2π radians = 1 rotation = 360o
1 rad = 360/2π = 57.296o
Often c used in Maths but use “rad” in Physics
1 radian is also the angle which subtends an
arc of a circle of length r, the radius
Centripetal acceleration
• The change in velocity = acceleration
time
• Acceleration is perpendicular to motion
• Speed is unchanged – only changes direction
• Acceleration is always towards the centre of
the circle
• It is called the
Centripetal Acceleration
Centripetal acceleration
•
•
•
•
Change in velocity v  v  v
'
Change in position r  r  r
But if  is small r  r
Also v  v
v r


v
r
r
But
v
t
V
v2

a
  2r
t
r
'
Centripetal force
Force = mass x acceleration
F  ma
( F  ma)
mv 2
F 
 m 2 r
r
Force is in the direction
of the acceleration
On the road – bridges and hills
mv 2
F
 mg  R
r
If R=0 so only just holding road
2
mv
 mg
r
 vmax  gr
Path of car
v  vmax
v  vmax
v  vmax
R
mg
On the road - roundabouts
• Friction provides centripetal force
2
mv
F friction 
r
2
mvmax
 Fmax  friction 
r
( Fmax  friction  mg )
• If frictional force not enough,
car continues in a straight line
or curved path – not sideways!
On the road - banking
• Banking provides a natural
force – cars move in a circle!
• Centripetal force
mv2
Fc 
 R sin 
r
• Reaction force
mg
R
cos 
sin 
 Fc  mg
 mg. tan 
cos 
At the fairground – the big dipper
• At the bottom of the dip
mv 2
Fn  mg 
r
mv2
 Fn  Support force 
 mg
r
• In this case...
if m  500kg
500  20
Fn 
 500  9.81  24900 N
10
2
At the fairground – the long swing
• Like “Rush” at Thorpe Park
• Potential energy
E p  mgl
1 2
 mgl  mv
2
• Kinetic energy
1 2
v  2 gl
Ek  mv
2
2
mv
and Ek  E p Also S  mg  l
 S  3mg
At the fairground – the big wheel
• At the top
mv 2
mg  R 
r
mv 2
R 
 mg
r
 v02  gr : R  0
• No force on
rider due to
chair
• At the bottom
2
mv
mg  R 
r
2
mv
R 
 mg
r
• Maximum force on
chair so feels heaviest
Ride safety
• Rides regularly checked for safety
• Incidents investigated by HSE – Health and Safety
Executive
• G-forces experienced by riders
• Max 1.9g into head
• Max 5.1g into feet
• Maximum g-forces on top
are at the back on Stealth
• Calculations for different sized
• People
Orbits
•
•
•
•
Real orbits
Ellipses
Sun is at a focus
Nearer the sun less Potential so more Kinetic
energy
• Force not completely perpendicular
• Speeds up and slows down as well as changing
direction
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