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Chapter 5

- Uniform Circular Motion
- acv2/r

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

2) Centripetal Acceleration

v

s

r

ac

- Directed toward centre
- Magnitude

3) Centripetal force

v

s

r

ac

Fc

- Toward centre (parallel to acceleration) since F

ma - Magnitude

4) Driving around circular curves

- a) Friction only

b) Banked curve, no friction

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

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

4) Driving around circular curves

- c) Banked curve with friction

Example Find maximum velocity for µ 0.80, ??

47º, and r 60 m

Fc

5) Satellites in circular orbits

- a) Speed and radius

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

5) Satellites in circular orbits

- b) Period, Synchronous orbits
- (i) Period and radius

Time for one revolution T

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

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.

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

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

and hit your head. The simulated gravity does not

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.

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)

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.

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.

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.

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.

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

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

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?

6) Centrifugal force and artificial gravity

This animation shows what happens when a ball is

dropped inside a rotating cylinder.

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

cylinder radii.

An object released moves with constant velocity

on the tangent following to Newtons first law.

6) Centrifugal force and artificial gravity

To an observer inside the cylinder, it moves up

and down (except for small displacement due to

finite radius of the cylinder).

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

return to it again very close the point of

departure.

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.

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

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

7) Vertical circular motion

7) Vertical circular motion

Minimum speed at top

For r 6 m, v 7.6 m/s or 27 km/h