Circular Motion

Why aren’t we getting anywhere?

AXIS OF ROTATION

This is an imaginary line through the center of the spinning

The points on the axis of rotation don’t move due to the spinning

As the circle spins the point on the axis of rotation appears is still

As the circle spins the point on the axis of rotation appears is still

Here the axis of rotation is not in the center, where is it?

If the axis of rotation is through the object it is called ROTATION

If the axis of rotation is outside of the object it is called REVOLVING

The earth rotates and revolves

The earth rotates and revolves

The earth rotates and revolvesAt the same time

The time it takes an object to complete 1 cycle is called the Period, its symbol is T

T

What is the unit?

Does the object have a constant speed?Does the object have a constant velocity?

Moving in a circle at a constant speed is called uniform circular motion

The speed of an object in a circular path is called its

Tangential Speed, vt

Its direction is always tangent to the circular path

How could we calculate the balls speed?

distancetime

vt = 2r T

For circular motion :

period (s)

radius (m)

tangential speed (m/s)

Which lady bug has the greater tangential speed?

The outer dot has to move faster in order to travel the longer path in the same amount of time

Does the object (at constant speed) accelerate?

Acceleration

a =v2 – v1

tWhy is this a bit harder here?

v1

v2

-v1

v2

The acceleration which keeps an object in

uniform circular motion is called-

Centripetal Acceleration aC

centripetal meanscenter seeking

Centripetal acceleration is always pointed to the center of rotation and perpendicular to the tangential velocity

aC

The magnitude is given by the equation below.

ac =vt

2

r

tangential velocity(m/s)

radius(m)

The faster it is going, the harder it needs to turn

The smaller the radius, the harder it needs to turn a sharp turn vs. a wide turn

The drum of a washing machine has a radius of 34 cm. During spin cycle the drum rotates with a tangential velocity of 5.5 m/s.

a.) What is the centripetal acceleration of the clothes against the drum?

b.) What is its direction?

c.) What is the period of rotation?

Two race cars are in a turn,the speedometer of 1 car reads a constant 60 mph and the second reads 120 mph. Compare their centripetal accelerations?

The speedometer of a car reads a constant 100 mph, the radius of the second turn is ½ of that of the first. Compare the centripetal accelerations between the turns.

Which penny has the greater acceleration?

acentripetal =v2

r

Compare the r’s

Compare the v’s

.1 m .2 m

2 pennies on a lazy susan demo

Since

acentripetal =v2

r

v = 2r T

=

4r2

r T2

ac =4r

T2

acentripetal =4r

T2

Compare their accelerations

T = 1 s T = 2 s

Is there a net force the object?What would happen to the ball if there was no net force acting on it?What is the direction of the net force on the object?

The force which cause an object to travel in uniform circular motion is called the CENTRIPETAL FORCE

It always points to the axis of rotation

A centripetal force is NOT a “new” type of force like FN, FF, FG, FT

It is simple one of the above acting in such a way to make an object travel in a circle.

But how does it “Know” to always point to the center, it must be COMPLICATED???

ball on a string demo

What is the centripetal force?

How does it always point to the middle?

The centripetal force is always perpendicular to the velocity!

What provides the centripetal force in the scenario

Fg

Note the centripetal force always points directly towards and perpendicular to the axis of “rotation”

What provides the centripetal force in the scenarioThat keeps the riders moving in a circle?

FN

What provides the centripetal force on the penny that keeps in moving in a circle on a record?

FF

What forces do you experience on a merry go round as it spins?

It feels like a force is pushing you to the outside of the circle, what is the dealer of this force?

Fg

FNFN

If the string were to be cut what path would the ball take?B

A C

D

E

F

Assume there is NO gravity

Anything curved has to go if no force is acting on the ball

AB

C

D

E

F

Anything curved has to go if no force is acting on the ball

AB

D

Let’s try it and find out?

What direction was the ball headed just before the string broke

AB

D

What direction was the ball headed just before the string broke

D

An overhead view of a car on a circular track, what provides the centripetal force

What happens if friction is lost?

The only (net) force in uniform circular motion is directed inward

Centripetal force

(tension)

There is NO outward directed force it only feels like it.

Tangential Velocity

The tangential velocity is the velocity is would continue to have if there were no net forces acting on it

Force

Does its speed change?

FC

The ball’s speed and KE never change,this means the force doesn’t do any WORK on it. WHY?The force is always perpendicular to its motion

F = ma

Fcentripetal = macentripetal

acentripetal =v2

r

Compare the centripetal force needed, equal mass

T = 1 s T = 2 s

acentripetal =v2

r

Which requires a greater Force?

The velocities are equal

When the record is spun faster which penny will fly off first?

.1 m .2 m

2 pennies on a lazy susan demo

acentripetal =4r

T2

acentripetal =4r

T2

Which requires a greater Force?

Which requires a greater Force?

acentripetal =4r

T2

Think back to the two pennies

Which has a bigger aC

Which has a bigger FC

A 1.00 kg ball is being swung around in a circle with a radius of .500 m by a string (in space). The ball completes a circle every 1.34 seconds. What is the :

tangential velocity of the ball?centripetal acceleration of the ball?force of tension on the string?

.40 m

A tetherball (1.5 kg) is swung in a circle with a tangential speed of .85 m/s, what angle does the rope make to the pole?

carousel applet

A 4.5 g penny is placed on a record, 15 cm away from the center. The coefficient of friction between the penny and the record is .25, what is the fastest tangential speed the penny can have before flying off.

In the ROTOR ride on a midway, passengers are pressed against the inside vertical wall of a rotating drum 3.5 m in radius. If it rotates at 40 rpm ...

How many g’s does a person experience?What normal force does a 70 kg person experience on the ride?What coefficient of friction will keep a person on the wall from sliding when the floor drops?

In a roto-ride WHY do you “feel” pressed against the wall, if there is NO actual force which does this?

Examples: going for a ride in a car-speed up, slowing down, making a right turn

Why do SCI FI writers often use wheel like rotating space stations?

If you were in a rotating spaceship, with No Gravity. Would a centripetal force be needed for you to travel in a circle?

YES!!!!!!!!!!!! Otherwise

If you were in a rotating spaceship, with No Gravity. Would a centripetal force be needed for you to travel in a circle?

YES!!!!!!!!!!!! Otherwise

What provides the centripetal force, & which way is it pointed?

From your perspective what direction would this normal force point?

The floor would always be pushing “UP”

Is there something pushing you “DOWN”

NO but it would FEEL like it. ARTIFICIAL GRAVITY!!!!!

How could you increase the artificial gravity on this spaceship?

Compare the artificial gravity felt by each person

ac=42rT2

ac=vt

2

r

Compare the gravity felt by your head to that felt by your feet..

ac=42rT2

ac=vt

2

r

If you were to jump off the ground, what forces would be acting on you? If you jumped up in such a space station, would you come back to the ground?

You would come back the ground if you jumped straight up. But you would simply be traveling in a straight line (diagonally) to meet with it

Jump up and hit the ground sideways

You could do some things that would seem strange, if you threw a ball opposite its tangential velocity (such that from an absolute frame of reference it was stationary. It would just sit there.

How fast would a person in a rotating spaceship need to be moving to experience an earth like feeling of gravity if the spaceship had a radius of 500 m?

What is weight?

On earth when do you “feel” heavier or lighter

What does your bathroom scale measure?

Mass or Force

Weight is a force

It changes depending on where you areearth vs. the moon

The FEELING of weight or weightlessness (like that tickle in your stomach when you go over a hill)comes from feeling the FORCES at work you.

Forces on your outside- like your butt in a chair or feet on the floor

Forces on your inside- your guts being pushed by your pelvic

bone

Your apparent “weight” also depends on if you’re movement

How do you feel when?

The elevator starts upward Heavier

after you get going Same

Slowing down to a stop Lighter

the elevator cord is cut Weightless

stationaryFN,you,floor

Fg,you,earth

you feel “normal”F net= 0

accelerating up @ 9.8 m/s2

In the example above you would “feel” twice as heavy, even though your weight is the same.

FN,you,floor

Fg,you,earth

F net=

stationaryFN,you,floor

Fg,you,earth

you feel “normal”F net= 0

moving up but slowing

You feel “lighter” not because you weigh more but because the floor is barely pushing on you. You get a tickle in your “stomach”

FN,you,floor

Fg,you,earth

F net=

stationaryFN,you,floor

Fg,you,earth

you feel “normal”F net= 0

Free fall

You feel weightless, are you?

FN,you,floor

Fg,you,earth

F net=

Just the right amount of centripetal force is needed to keep an object moving in a circle.

In many cases the applied force will always match the right amount

Swinging a ball on a string

What happens if the ball is swung faster?

Just the right amount of force centripetal force

FC = mac

This is just the right amount (your answer)

When an object is swung in a vertical circleThe FNET between FG & FT supply the centripetal force

FC

FG

At the bottom of the swing, the string has to pull hardest to provide a Net force to the middle of the circle

FC

FG

FCFT

At the TOP of the swing, the string pulls the least because gravity is almost enough

FC

FGFC

FT

What happens to the needed centripetal force if it is swung slower?

FC

FGFC

FT

A 3.5 kg ball is cleverly swung in a vertical circle at a constant of 4.5 m/s on a .56 m string. What is the tension in the string at the top and bottom positions?

If too much centripetal force is applied then the object will

If not enough centripetal force is applied then the object will

Now gravity is providing more centripetal force than is needed, what happens?

FC(needed)

FG

FC (needed)

Swinging water demo

If this space station is placed on earth as the highland fling, what you feel changes based on where you are in the rotation

If it is just spinning fast enough to keep you from falling off

You spin just fast enough so that the Fg is just enough ( or =) centripetal force needed. At the top you feel….

Fg

Fc

(At the same speed). At the bottom, the floor has to push twice as hard as your weight

And you feel….

Fg

Fc

FN

COASTER APPLET

When does the coaster fall off the top of the loop?

Why are roads banked around turns instead of flat?

FN

Fg

FF

FN

Fg

FF

In what direction is a force needed to get the car around a turn?(the centripetal force). What direction is the axis of rotation?

Fg

FN

centripetal force needed

When the roads are banked, a component of the normal force acts in the direction of centripetal force. (lowering the need for friction).

How fast can a car take a turn at 23 m/s on frictionless ice without skidding on the turn if that road is banked at 8o ?

Fg

FN

centripetal force needed

What is the direction of the acceleration?

Note that here standard axes work, don’t rotate!!!!

What does centripetal acceleration do?

What would tangential acceleration do?

If a ball is in uniform circular motion it accelerates, but its tangential velocity doesn’t change.

Why doesn’t the force cause the object to speed up or slow down here? It only changes direction.

ac

vt

A centripetal force is always perpendicular to velocity. So it just causes a change in direction.

ac

vt

Often an object speeds up or slows as it turns. Its tangential velocity changes due to tangential acceleration.

ac

at

ac

at

ac

ac

Why does its centripetal acceleration get more and less?

ac

at

ac

at

ac

ac

lets pretend it doesn’t stop at the top.

What causes the tangential acceleration here?

ac

at

ac

at

ac

ac

Why is there no tangential acceleration at the top and bottom?

ac

at

ac

at

ac

ac

Finish first problem set

A brief history of gravity & the planets

Aristotle contributed many theories to science that were believed for many years such as:

the natural state of an object is to be at rest

the four elements

and the earth is the center of the solar system (geocentric)

Why was it so hard to figure out that the sun was the center?

It is a matter of perspective:

mars retrograde motion movie

The geocentric view, that every planet, star, and moon revolved around the earth. And it had some problems:

These moons are not revolving around the earth,

Perhaps the earth is not the center of the universe?

The catholic church did not take kindly to his idea

Galileo recants and is spared but placedunder house arrest for the rest of his life.

But the Catholic Church did sort of apologize

In 1979 Pope John Paul II said that Galileo may have had divine inspiration

Tycho BraheTI-KO Bra

Took meticulous measurements of the movements of the planets and stars around 1600

BUT, he believed in the geocentric model.

Brahe, got part of his nose cut off with a swordin a duel.

He had a prosthetic made of metal which he would glue on.

Tycho Brahe had an apprentice,

Johannes Kepler

Believed in the sun-centered view of the solar system

From his analysis Kepler established:

3 LAWS of PLANETARY MOTION

Kepler’s Laws of Planetary Motion

1) The paths of the planets are ellipses with the sun at one focus.

Kepler’s Laws of Planetary Motion

2) Planets sweep out equal areas in equal times.

elliptical orbit animation

Kepler’s Laws of Planetary Motion

3) The ratio of the squares of the periods of two bodies orbiting a third is equal to the ratio of the average radii cubed.

(Ta/Tb)2 = (ra/rb)3

1 AU = 93 million miles

An asteroid orbits the sun at 2 AU’s. What is its period?

a) 2.0 years

b) 8.0 years

c) 2.8 years

d) 5.0 years

An asteroid orbits the sun at 2 AU’s. What is its period?

a) 2.0 years

b) 8.0 years

c) 2.8 years

d) 5.0 years

Why are Kepler’s Laws only LAWS not theories

because they only predict what will happen not WHY

The key was to understand what force causes these planets to travel in these paths?

Are you attracted to the person sitting next to you?

YES!!

ALL matter experiences an attractive force to every other bit of matter. It’s called

GRAVITY

Beside creating the 3 Laws of motion andinventing Calculus:

Isaac Newton also taught us a lot about gravity and the motion of planets

WHAT A STUD!

After being hit with an apple, his thoughts led him to wonder what kept the moon in orbit. The answer…..

Gravity

Newton knew that the moon’s velocity changed constantly and therefore there must be a force on it.

Might the earth pull on the moon just like the apple?

But the apple is much closer to the earth than the moon.

The apple is a distance 1 earth radius from the center of mass of the earth

The moon is 60 earth radii away.The apple is only 1.

Note: apple not to scale

Newton calculated that the amoon wasof the acceleration of the apple

13,600

if distance increased by 60, the acceleration decreased by 3,600. Newton concluded that the force of gravity must decrease by distance2.

or1

602

So far

Fg =something

r2

Newton further reasoned, that since a 2 kg object was twice as heavy as a 1 kg object.

The force of gravity must be directly proportional to the mass of the object being measured.

So far

Fg =mobject

r2

Newton thought back to his own 3rd law.

When the apple feels a pull from the earth.

The earth feels a pull from the apple.So the earth’s mass must be a factor as well.

Fg = Gm1 m2

r2

r (m)

m1 (kg) m2 (kg)

A fudge factor(Newton was unable to determine this #)

Force on either object due to gravitational attraction

Newton’s Law of Gravity

Fg = Gm1 m2

r2

Fg = Gm1 m2

r2

Newton believed ALL objects with mass experience a gravitational attraction.

Why not just take a couple known masses, set them some distance apart, measure the force between them and do the victory dance?

Fg = Gm1 m2

r2

Objects that could be weighed exerted extremely small attractive forces.

Objects which exerts large forces (like the earth are hard to weigh)

Henry Cavendish

English chemist and physicist who was shy and absent-minded. He was terrified of women, and communicated with his female servants by notes.

9.8 N =1kg mearth

(6,380,000 m)2

G

A 1 kg object on the surface of the earth has a force of gravity of 9.8 N between it and the earth. The radius of

the earth is 6,380,000 m.

With G known, they could find the mass of the earth

mearth = 5.98 1024 kg

Cavendish’s experiment is often said to have weighed the earth in his

experiment

5.98 1024

Newton’s Law of Gravity

Every particle is attracted to every other particle with a force that is proportional to the product of the 2 masses divided by the square of the distance between them

Fg = Gm1 m2

r2

Universal Gravitional Constant(fudge factor)= 6.67 x 10-11 N m2

kg2

distance between objects (m)

mass of objects (kg)

Why does the force of Gravity weaken exponentially ( 1/r2 ) with distance.

When a light shines it spreads out as a spheregetting dimmer as you go further

Picture a hollow sphere with the light bulb in the middle.

If the sphere was doubled in radius, the same amount of light is spread out over 4 times the area.

Area of a sphere = 4 r2

So it is 4 times dimmer

Picture a butter gun

Many things follow the INVERSE SQUARE law:Gravity, Light, Sound, Magnets, Radioactivity……

1 m2 m

3 m

It shoots out the same amount of butter each time

The further away, the more it can cover with 1 shot

1 inch thick

Earthm= 5.98x1024 kg

Moonm= 7.35x1022 kg

What is the force between them?

=mearth mmoon

r = 3.84x108

r2GFg

What is the Fg between the earth and the moon

Earth Moon

Which object feels a greater FORCE of gravity?

Fg,earth,moon = Fg,moon,earth

r

=mearth mmoon

r2G

Earthm= 5.98x1024 kg

Moonm= 7.35x1022 kg

What is the force between them?

=mearth mmoon

r = 3.84x108

r2GFg

What would weaken the attraction between them more:

Halving the mass of the earthHalving the mass of the moondoubling the distance between them

Force of gravity between you and your neighbor?Say you both have a mass ~ 70 kg, and are a distance of 50 cm apart.

What force does an object feel at the surface of the earth, if the object has a mass = mobject. What is its acceleration?

mearth= 5.98x1024 kgrearth= 6.38x106 m

rearth

1st: why use the radius of the earth as the distance?

2nd: what is the force felt

The radius of the earth is 6,380,000 m.

The “weightless” astronauts orbit the earth at 200,000 m above it surface. What is the

acceleration due to gravity at that altitude?

Gravitational pull is still 94% of what it is on earth, why are they weightless?

If you weigh 100 N on the earth’s surface, what would your weight be on a ladder equal to the radius of the earth?

rearth

To what point on the earth are you attracted to?

WHY?

How would the force of gravity change if you were somehow inside the earth?

What would happen to the force of gravity on you if you jumped in a big hole through the earth?

What would happen to the earth, if the sun collapsed into a black hole?

Nothing, except we would freeze!

Newton showed us that the same laws of physics on earth apply to the rest of the

universe

Two galaxies crashing

What is concerning about the fact that everything in the universe attracts

everything else?

Einstein’s Theory of Gravity

• Gravity is not a force.

• It is an effect of curved space-time.

Gravitational lensing

Gravitational lensing animation

The Night Sky

With a bit better view

Gravity Problems

Honors Physicspage 141 25, 26, 31, 34, 52

An object will orbit a planet when the force of gravity is equal to….

Fcentripetal

V

Fg

Shooting cannon balls

If a cannonball is shot relatively slowly

gravity pulls it down quickly to the surface

Fg>Fc

Shooting cannon ballsShooting cannon balls

If a cannonball is shot a bit faster

it goes further but is still brought down

Fg>Fc

Shooting cannon ballsShooting cannon balls

If a cannonball is shot much faster

it escapes the planets gravity

Fg<Fc

Shooting cannon ballsShooting cannon balls

Somewhere in between the last 2 shots

it continues to fall, but it tangential velocity keeps it from ever hitting the planet. It is in orbit.

Fg=Fc

How fast must a satellite be moving if its mass is 10.0 kg and it is 7.5 x 109 m (from the center of the earth)?

FC = mv2

r Fg = m1 m2

r2G

Mass of satellite does not matter for orbitonly factors are : distance, speed, mass of central planet or sun

FC = m1

v2

r Fg = m1 m2

r2G

mass of satellite

central mass--planet or sun etc...

FC = Fg

m1v2

r = G

m1 m2

r2

v2 = G m2

r

central mass

For an object in orbit

v2 = G mr

v = 2r

T

central mass

What is the tangential velocity of an object moving in a circle?

2r T

G mr=

2

4r2 T2

G mr=

4r3 = G m T2

The cube of radius is proportional to the square of period

cross multiplying

4ra3 = G m Ta

2

4rb3 = G m Tb

2

For two objects orbiting the same object. A and B.

4ra3 = G m Ta

2

4rb3 = G m Tb

2

Comparing the orbits of two satellitesA & Bby dividing the two equations:

ra3 Ta

2

rb3 Tb

2=

Ta da:Kepler’s 3rd law.ALL OF THESE equationswork for circular or elliptical orbits

What distance from the center of the earth would a satellite have to be in order to remain above the same location the earth.

T = 1 day

2r T

G mr=

2Which equation to use?

What is given

mearth = 5.98x1024 kg

G = 6.67x10-11

r = ???

Consider two satellites in orbit about a star (like our sun). If one satellite is twice as far from the star as the other, but both

satellites are attracted to the star with the same gravitational force, how do the masses of the satellites compare?