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## Uniform Circular Motion

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### 5) Satellites in circular orbits. a) Speed and radius ... 5) Satellites in circular orbits. b) Period, Synchronous orbits (ii) Radius of synchronous orbit: ... – PowerPoint PPT presentation

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Title: Uniform Circular Motion

1
Chapter 5
• Uniform Circular Motion
• acv2/r

2
Uniform circular motion
Motion in a circular path with constant speed
v
s
r
1) Speed and period
• Period, T time for one revolution
• Speed is related to period
• Path for one revolution s 2?r
• Speed v s/T 2?r/T

3
2) Centripetal Acceleration
v
s
r
ac
• Directed toward centre
• Magnitude

4
3) Centripetal force
v
s
r
ac
Fc
• Toward centre (parallel to acceleration) since F
ma
• Magnitude

5
4) Driving around circular curves
• a) Friction only

b) Banked curve, no friction
6
4) Driving around circular curves
• c) Banked curve with friction

Net force F is vector sum of gravity, the normal
force, and friction
fs(ii)
Since the ideal (zero friction) angle is given by
fs(i)
i) If v2 gt gr tan ???friction prevents sliding up
ii) If v2 lt gr tan ???friction prevents sliding
down
7
4) Driving around circular curves
• c) Banked curve with friction

Example Find maximum velocity for µ 0.80, ??
47º, and r 60 m
Fc
y-motion
x-motion
8
4) Driving around circular curves
• c) Banked curve with friction

Example Find maximum velocity for µ 0.80, ??
47º, and r 60 m
Fc
9
5) Satellites in circular orbits

Radius of orbit determines speed (independent
of mass) Accelerations of all objects at the
same radius are equal (no acceleration relative
to each other) gt No apparent force between
them apparent weightlessness in orbiting
satellite
10
5) Satellites in circular orbits
• b) Period, Synchronous orbits

Time for one revolution T
11
5) Satellites in circular orbits
• b) Period, Synchronous orbits
• (ii) Radius of synchronous orbit
• Definition satellite is stationary above
earths surface
• Conditions  T 1 sidereal day
• above equator

12
6) Centrifugal force and artificial gravity
• Artificial gravity
• A rotating cylinder exerts a centripetal force
on objects on the inside surface e.g. if r
1700 m, and Fc mg (usual force of gravity),
then mg mv2/r, giving

Centripetal force acts toward the centre (up,
like the normal force on earth) but gravity acts
down. What has taken the place of gravity, from
the perspective of the cylinder-dwellers?
Centrifugal force does not exist is a
fictitious, pseudo, virtual force is an inertial
force
The concept of centrifugal force is not
required material.
13
6) Centrifugal force and artificial gravity
• Artificial gravity
• A rotating cylinder exerts a centripetal force
on objects on the inside surface e.g. if r
1700 m, and Fc mg (usual force of gravity),
then mg mv2/r, giving

Centripetal force acts toward the centre (up,
like the normal force on earth) but gravity acts
down. What has taken the place of gravity, from
the perspective of the cylinder-dwellers?
Centrifugal force does not exist (in inertial
frames) is a fictitious, pseudo, virtual
force is an inertial force
14
6) Centrifugal force and artificial gravity
• This episode of Quirks Quarks on CBC radio
illustrates the confusion that can exist about
artificial gravity.
• http//www.cbc.ca/quirks/archives/01-02/mp3/qq13
1001f.mp3

The question was What happens when you jump
inside a rotating cylinder like the one on the
movie 2001 A Space Odysey Contrary to the
answer given, you do not float to the other side
require continuous contact with the surface. From
your perspective (the jumpers), you will be
pulled back to the surface just as you would in a
gravitational field, apart from a small deviation
depending on the radius of the cylinder. To
understand this, we will consider a simpler
inertial force first.
15
6) Centrifugal force and artificial gravity
• b) Inertial force -- Apparent force resulting
from an accelerating reference frame.
• e.g. An accelerating spaceship (far from planets)

16
6) Centrifugal force and artificial gravity
• b) Inertial force -- Apparent force resulting
from an acceleration reference frame.
• e.g. An accelerating spaceship (far from planets)

m
a0
ma0
FN
Inside the spaceship, acceleration is zero, so
net force should be zero. Same equation, new
interpretation Inside the ship an inertial force
equal to ma0 is acting toward the floor.
17
6) Centrifugal force and artificial gravity
• b) Inertial force -- Apparent force resulting
from an acceleration reference frame.
• e.g. An accelerating spaceship (far from planets)

a0
Inside the spaceship, acceleration is zero, so
net force should be zero. Same equation, new
interpretation Inside the ship an inertial force
equal to ma0 is acting toward the floor. A
dropped ball falls to the floor just as it would
if a constant force acted on it.
18
6) Centrifugal force and artificial gravity
• b) Inertial force -- Apparent force resulting
from an acceleration reference frame.
• e.g. An accelerating spaceship (far from planets)

a0
Inside the spaceship, acceleration is zero, so
net force should be zero. Same equation, new
interpretation Inside the ship an inertial force
equal to ma0 is acting toward the floor. A
dropped ball falls to the floor just as it would
if a constant force acted on it.
19
6) Centrifugal force and artificial gravity
• b) Inertial force -- Apparent force resulting
from an acceleration reference frame.
• e.g. An accelerating spaceship (far from planets)

a0
Inside the spaceship, acceleration is zero, so
net force should be zero. Same equation, new
interpretation Inside the ship an inertial force
equal to ma0 is acting toward the floor. A
dropped ball falls to the floor just as it would
if a constant force acted on it.
20
6) Centrifugal force and artificial gravity
• c) Centrifugal force the inertial force in a
rotating reference frame

Consider a cup of water swung around in a
vertical circle. What keeps the water in the cup
when it is upside-down?
In an inertial frame, only the normal force and
gravity act to give centripetal force which
produces centripetal acceleration (no such thing
as centrifugal force)
Answer Inertia. The cup pulls the water down
faster than its natural falling rate.
Gravity plus the normal force provide centripetal
force that produces circular motion
21
6) Centrifugal force and artificial gravity
• c) Centrifugal force the inertial force in a
rotating reference frame

Inside the cup, the water is at rest, as is an
observer in a boat inside the cup i.e. they do
not accelerate in this reference frame.
What keeps water in the cup from the perspective
of the cup-dweller?
Answer Invent centrifugal force to balance real
forces and ensure zero acceleration. So
22
6) Centrifugal force and artificial gravity
• c) Centrifugal force the inertial force in a
rotating reference frame

What happens if a ball is dropped?
If the force holding it Fh is removed, it should
fall radially (down to the cup-dweller), but in
an inertial frame it moves tangentially.
Fh
How can both be true?
23
6) Centrifugal force and artificial gravity
This animation shows what happens when a ball is
dropped inside a rotating cylinder.
24
6) Centrifugal force and artificial gravity
In the rotating reference frame it moves radially
outward (down to the observer), apart from a
small curvature which decreases for larger
An object released moves with constant velocity
on the tangent following to Newtons first law.
25
6) Centrifugal force and artificial gravity
To an observer inside the cylinder, it moves up
and down (except for small displacement due to
An object that leaves the floor with a small
radial velocity (like a jumping person) will have
velocity with both tangential and radial
components and will move at a small angle to the
the tangent. It will leave the floor and then
departure.
26
6) Centrifugal force and artificial gravity
• c) Centrifugal force the inertial force in a
rotating reference frame

An accelerating reference frame can duplicate the
effect of gravity (not only when in contact).
Centrifugal force is introduced within the
reference frame to account for the observed
acceleration.
However, when analyzing the situation from an
inertial frame (as we do in this course),
inertial forces are not present. Centrifugal
force does not exist.
27
6) Centrifugal force and artificial gravity
• d) Equivalence principle
• - Inertial mass (F ma) is the same as
gravitational mass (F GmM/r2)
• - It is, however, clear that science is fully
justified in assigning such a numerical equality
only after this numerical equality is reduced to
an equality of the real nature of the two
concepts. -- Einstein
• - The happiest thought of my life
• The gravitational field has only a relative
existence... Because for an observer freely
falling from the roof of a house - at least in
his immediate surroundings - there exists no
gravitational field. -- Einstein

A uniform gravitational field is completely
equivalent to a uniformly accelerated reference
frame. No local experiment can distinguish
them Gravity is geometrical
28
7) Vertical circular motion
• Not usually uniform motion since the speed is
changing
• Net force not always toward the centre
• Component of acceleration toward the centre (the
centripetal component) is still v2/r, so

29
7) Vertical circular motion
30
7) Vertical circular motion
Minimum speed at top
For r 6 m, v 7.6 m/s or 27 km/h