Uniform Circular Motion and Uniform Circular Motion and GravitationGravitation

Yet another reason why Star Wars is fictitious…

Features of UCMFeatures of UCM

• Constant speed

• “center-seeking” acceleration

• Tangential velocity

Rotation with Rigid ObjectsRotation with Rigid Objects

• For our purposes, we’ll talk about rigid objects

• Rigid objects have the same rotational period at all points

• Ex: merry-go round, CD

Exceptions?Exceptions?

• http://upload.wikimedia.org/wikipedia/commons/a/a3/790106-0203_Voyager_58M_to_31M_reduced.gif

Imagine sitting 0.5 m from the center of Imagine sitting 0.5 m from the center of a merry-go-round. How does your a merry-go-round. How does your speed change if you double your speed change if you double your

distance from the center? distance from the center?

• 1. It doubles

• 2. It decreases by half

• 3. It remains the same

• 4. It increases by a factor of 4

• 5. It decreases by 1/4

What happens to your acceleration if What happens to your acceleration if you double your distance from the you double your distance from the

center of the merry-go-round?center of the merry-go-round?

• 1. It doubles

• 2. It decreases by half

• 3. It remains the same

• 4. It increases by a factor of 4

• 5. It decreases by 1/4

As a car rounds a corner, which As a car rounds a corner, which wheels are spinning faster, the wheels are spinning faster, the

wheels on the inside of the turn (1) wheels on the inside of the turn (1) or the outside (2)? or the outside (2)?

Bonus: how can they do this if Bonus: how can they do this if attached to the same axle? attached to the same axle?

Demos/ApplicationsDemos/Applications

• Rail road tracks

• CD tracking:

• http://electronics.howstuffworks.com/cd6.htm

The Driving Force of UCMThe Driving Force of UCM

• Think about twirling a tennis ball above your head with a rope

• In which direction do you have to pull to keep the tennis ball in a uniform circular path?

• A center-seeking force is called a centripetal force

What is it? What is it?

• The centripetal force is similar to the net force (in fact, it is a type of net force)

• It is the result of the external forces acting upon an object

• Centripetal force is not a separate force like gravity or friction

• Fc is caused by external forces like gravity

Newton’s 2Newton’s 2ndnd, Remixed, Remixed

• Fnet = ma

• For circular motion:

• Fc = mac = m(vT2/R)

A tennis ball has a mass of A tennis ball has a mass of 0.060kg. What tension force is 0.060kg. What tension force is

required to swing the ball around in required to swing the ball around in a circular path of radius 0.5m at a a circular path of radius 0.5m at a

speed of 5m/s? speed of 5m/s?

A car (m = 2000kg) is about to round a A car (m = 2000kg) is about to round a curve of radius 20m. Assuming the curve of radius 20m. Assuming the

coefficient of friction between the road coefficient of friction between the road and tires is 0.7, what is the maximum and tires is 0.7, what is the maximum

speed the car can round the curve speed the car can round the curve without slipping? without slipping?

Centrifugal “Forces”Centrifugal “Forces”

• Say you’re in a car, rounding a curve

• What does it feel like?

• We sometimes call this sensation a centrifugal force

• What is the origin of this force?

It’s All About PerspectiveIt’s All About Perspective

• From inside the car, it appears that you are being pulled upon by some imaginary “force”

• From above, what are you doing?

• Following Newton’s first law

• Centrifugal forces are not forces at all

• They are a by-product of our inertia when placed in accelerated situations

• This is an example of the dangers that occur when analyzing physics in an accelerated situation (reference frame)

ExamplesExamples

• Centrifuges

• Amusement park rides

• Washing machine

• So as much as I love Star Wars, I can’t help but notice that people tend to walk normally while riding in space ships

• What’s wrong with this picture?

Artificial GravityArtificial Gravity

• How could we use rotation to create “artificial” gravity in space?

• We could use the centrifugal effect to produce a faux gravitational force

Vertical CirclesVertical Circles

So How Much Do You Weigh on So How Much Do You Weigh on the Sun? the Sun?

Universal GravitationUniversal Gravitation

• Gravity doesn’t stop at Earth’s boundaries

• It is the dominant force in the universe over large scales

• What are gravity’s origins?

• How do we calculate gravitational force?

Gravity’s OriginsGravity’s Origins

• Object’s with mass, separated by some distance R

• Fg = Gm1m2/R2

Say I have two objects, A and B, Say I have two objects, A and B, separated by distance R, which exert a separated by distance R, which exert a gravitational force F on one another. gravitational force F on one another.

By what factor does that force change if By what factor does that force change if they are moved to a distance of 2.5R they are moved to a distance of 2.5R

apart? apart?

Calculate the gravitational pull Calculate the gravitational pull between you and your neighbor. between you and your neighbor.

Acceleration due Acceleration dueto gravity?to gravity?

Who has a stronger Who has a stronger gravitational pull: a person gravitational pull: a person

beside you (with the same mass beside you (with the same mass as you), or the neighbor behind as you), or the neighbor behind you (twice as massive, but twice you (twice as massive, but twice

as far away). Assume 1 is for as far away). Assume 1 is for the neighbor beside you, while 2 the neighbor beside you, while 2

is for the neighbor acrossis for the neighbor across

• But I thought Fg = mg ?!

• Right you are…

• So if Fg = mg = Gm1m2/R2, we can write a new expression for g

g revisitedg revisited

• g = Gm2/R2 , which is sometimes called gravitational field strength

Say I travel to three celestial bodies: Say I travel to three celestial bodies: Earth, the moon, and Saturn. Rank Earth, the moon, and Saturn. Rank my weight in each place in increasing my weight in each place in increasing

orderorder

• 1. Earth, moon, Saturn

• 2. Earth, Saturn, moon

• 3. Moon, Saturn, Earth

• 4. Moon, Earth, Saturn

• 5. Saturn, Earth, Moon

Suppose I weigh 750N on Earth. Suppose I weigh 750N on Earth. What would I weigh on the Sun’s What would I weigh on the Sun’s

surface? (g = 275m/s/s)surface? (g = 275m/s/s)

g values around the solar systemg values around the solar system

• Earth’s Surface 9.8 m/s/s• Moon’s Surface 1.6 m/s/s• Mars’s Surface 3.8 m/s/s• Jupiter’s Surface 26.0 m/s/s• Sun’s Surface 275 m/s/s• Black Hole’s Event Horizon

4,000,000,000,000,000,000 m/s/s

• How much does a 80kg person weigh on Mt. Everest (h = 5.51 miles above sea level = 8.85km)?

• g = 9.79 m/s/s

• W = 783N (compared to 784N)

Black Hole MassBlack Hole Mass

• The event horizon of a “typical” black hole is about 10 km in radius

• Given the value for g (= 4E12 m/s/s) find the approximate mass of a “typical” black hole

• Give in solar masses (# of times the mass of our sun)

• 3 solar masses

Cosmic Vacuum Cleaners?Cosmic Vacuum Cleaners?

• Blackholes are sometimes thought of as large vacuums that suck everything into them

• From this point of view, the universe will ultimately be consumed by the current collection of black holes…

The Real DangerThe Real Danger

• Fortunately, this interpretation is wrong

• The real danger of a black hole lies in its

composition: you have a TONTON of mass stuffed into a very small space

• Consider this: a black hole is as heavy as our Sun, but is about 10 miles wide

• Our Sun is about 1 million miles wide

• Outside a black hole, the “tug” of gravity

would be about 10 billion times stronger than at the surface of our sun, or

about 1 Trillion times Earth’s gravity

Earth/Sun GravityEarth/Sun Gravity

• What is the force of gravity between the Earth and the sun

• The average distance between the two is 1.5 x 108 km

• How does this compare to the number we got last week with our circular motion calculation (3.55E22N)?

The Galactic CenterThe Galactic Center

• Astronomers have fairly good evidence of a super massive black hole in the galactic center

• How do they know?

• Obviously, we can’t “weigh” the galactic center on a scale

• However, we can use the concepts of circular motion and gravitation to indirectly measure the galactic center’s mass

Satellite MotionSatellite Motion

• An understanding of satellite motion will help…

• Let’s imagine objects orbiting Earth

• What does it mean to orbit the Earth?

• You’re traveling in either a circular or elliptical path around Earth

• Gravity is keeping you in that path

• For circular orbits, we’ve got UCM

• At what speed does the moon orbit the earth?

• Useful information: 384,000 km between the Earth’s center and the moon’s center

• 1020 m/s

• At what speed does a space shuttle orbit the Earth? (altitude = 300km)

• At what speed would it need to travel to orbit at the height of Mount Everest? (altitude = 8.85 km)

• 7700 m/s; 7900 m/s

Geosynchronous OrbitGeosynchronous Orbit

• Certain satellites are in what is called geosynchronous orbit

• They literally stay above one point on Earth’s surface

• What is their orbital radius?

• Orbital speed?

• A relationship between orbital period and orbital radius (for one or more objects)?

• This relationship is called Kepler’s Third Law

• Venus orbits the sun at a radius of 1.08 E 8 km

• How many days is one year on Venus?

Mass of Star XMass of Star X

• In an unknown solar system, astronomers observe a planet orbiting its parent star with a period of 200 days, at a distance of 0.5 AU

• What is the mass of the parent star?

The Galactic CenterThe Galactic Center

The Catch…The Catch…

• Our calculations involve uniform circular motion

• Turns out that these stars follow elliptical paths, meaning they travel at different speeds during different portions of their orbit

• The process is similar however…