Uniform Circular Motion

Physics 12

Centripetal Acceleration

• In order for an object to follow a circular path, a force needs to be applied in order to accelerate the object

• Although the magnitude of the velocity may remain constant, the direction of the velocity will be constantly changing

• As a result, this force will provide a centripetal acceleration towards the centre of the circular path

How can we calculate centripetal acceleration?

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Centripetal Force

• Like the centripetal acceleration, the centripetal force is always directed towards the centre of the circle

• The centripetal force can be calculated using Newton’s Second Law of Motion r

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Problem – horizontal circle

• A student attempts to spin a rubber stopper (m = 0.050kg) in a horizontal circle with a radius of 0.75m. If the stopper completes 2.5 revolutions every second, determine the following:– The centripetal acceleration– The centripetal force

• The stopper will cover a distance that is 2.5 times the circumference of the circle every second

• Determine the circumference

• Multiply by 2.5• Use the distance and

time (one second) to calculate the speed of the stopper

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• Use the speed and radius to determine the centripetal acceleration

• Then use the centripetal acceleration and mass to determine the centripetal force

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Problem – vertical circle• A student is on a carnival ride

that spins in a vertical circle.– Determine the minimum speed

that the ride must travel in order to keep the student safe if the radius of the ride is 3.5m.

– Determine the maximum force the student experiences during the ride (in terms of number of times the gravitational force)

Problem – vertical circle

Vertical Circle

• While travelling in a vertical circle, gravity must be considered in the solution

• While at the top of the circle, gravity acts towards the centre of the circle and provides some of the centripetal force

• While at the bottom of the circle, gravity acts away from the centre of the circle and the force applied to the object must overcome both gravity and provide the centripetal force

Vertical Circle

• To determine the minimum velocity required, use the centripetal force equal to the gravitational force (as any slower than this and the student would fall to the ground)

• To determine the maximum force the student experiences, consider the bottom of the ride when gravity must be overcome

• At the top of the circle, set the gravitational force (weight) equal to the centripetal force

• Solve for velocity

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• At the bottom of the circle, the net force is equal to the sum of the gravitational force and the centripetal force

• Solve for number of times the acceleration due to gravity

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Road Design

• You are responsible to determine the speed limit for a turn on the highway. The radius of the turn is 55m and the coefficient of static friction between the tires and the road is 0.90.– Find the maximum speed at which a vehicle can

safely navigate the turn– If the road is wet and the coefficient drops to 0.50,

how does this change the maximum speed

Diagrams

The maximum speed at which a vehicle can safely navigate the turn

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Coefficient drops to 0.50, how does this change the maximum speed

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The Motion of Planets

Birth of Modern AstronomyOR

How Nerds Changed the World!!!

• explain qualitatively Kepler’s first and �second laws and apply quantitatively Kepler’s third law

• explain and apply the law of universal gravitation to orbital notations by using appropriate numeric and graphic analysis �

• distinguish between scientific questions and technological problems as applied to orbital situations

Learning Outcomes (Students will be able to…):

Assumptions of Early Models of the Solar System (from the time of Aristotle…)

• Geocentric - Earth in the middle

• Everything orbits the Earth

• Stars are located on the Celestial Sphere

• Everything moves in uniform circular motions

Earth

Deferent

Epicycle

Mars

Equant

Claudius Ptolemy (87-165)

Nicolaus Copernicus (1473-1543)

•Errors building up

•Must be a better way!

•Let’s try a Heliocentric (or Sun-centered) system!

•Not any better though

Tycho Brahe (1546-1601)

•Comet – beyond the Moon

•Supernova – far away

•Naked eye observations of planets

•Accuracy through repetition

•Best observations of planetary positions

•Hired “nerd” to help calculate model

•Died….

Johannes Kepler (1571-1630)

•Worked for Brahe

•Took data after his death

•Spent years figuring out the motions of the planets

•Came up with…

Three Laws of Planetary Motion

1st Law: Planets move in elliptical orbits with the Sun at one foci

Sun

Foci (sing. Focus)

Perihelion Aphelion

Average distance from the Sun = 1 Astronomical Unit (1 A.U.) = approx. 150 000 000 km

2nd Law: Planets move faster at perihelion than at aphelion OR a planet sweeps out equal areas in equal time periods.

1 Month1 Month

3rd Law: Period is related to average distance

T = period of the orbit

r = average distance

kT2 = r3

•Longer orbits - greater average distance

•Need the value of k to use the formula

•k depends upon the situation

•Can be used for anything orbiting anything else

Special version of Kepler’s third Law –

If the object is orbiting the Sun

T – measured in years,

r – measured in A. U., then….

T2 = r3

For planets A and B, Kepler’s 3rd Law can look like this…

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Galileo Galilei (1564-1642)

•Knew of Copernicus’s & Kepler’s work

•Used a telescope to look at the sky

•What did he see?

The Moon was an imperfect object

Venus has phases

Jupiter has objects around it

Saturn is imperfect

The Sun is imperfect

Isaac Newton (1642-1727)

•The ultimate “nerd”

•Able to explain Kepler’s laws

•Had to start with the basics -

The Three Laws of Motion

1. Law of Inertia - Objects do whatever they are currently doing unless something messes around with them.

2. Force defined

F = ma

F=forcem=massa=acceleration (change in motion)

3. For every action there is an equal and opposite reaction.

The three laws of motion form the basis for the most important law of all (astronomically speaking)

Newton’s Universal Law of Gravitation

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F=force of gravity

G=constant (6.67 x 10-11 Nm2/kg2)

M1, M2 = masses

R=distance from “centers”

Gravity is the most important force in the Universe

An Inverse Square Law…

Newton’s Revisions to Kepler’s Laws

• Newton agreed with 1st law of motion

• Defined bound orbits (i.e. circular, elliptical) and unbound orbits (i.e. hyperbolic, parabolic) with Sun at one focus

• Used conic sections to describe orbits

Newton’s Revisions to Kepler’s Laws

• Newton agreed with 2nd law of motion

• Believed planetary motion to be non-constant acceleration due to varying distance between planet and Sun

• Force causing acceleration was gravity

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•4π2 and G are just constant #s (they don’t change)

•M1 and M2 are any two celestial bodies (could be a planet and Sun)

•Importance: if you know period and average distance of a planet, you can find mass of Sun (2 x 1030 kg)

or any planet!

Mass of Sun is 2 000 000 000 000 000 000 000 000 000 000 kg

Mass of Earth is 6 000 000 000 000 000 000 000 000 kg

Mass of Mr. J is 100 kg! WOW!

Newton’s Revisions to Kepler’s Laws•Newton

extended 3rd law to…

Newton’s Mountain

• Horizontal projectile launched at 8km/s

• How far does the projectile fall in one second?

• How far does the Earth “fall” away from the projectile?– Assume that arc length and chord length are

equal over the 8km distance and the Earth’s radius is 6400km

Newton’s Mountain• Shortly after developing the Universal Law of

Gravitation, Newton began a series of thought experiment involving artificial satellites

• Newton’s thought was that if you had a tall enough mountain and launched a cannonball fast enough horizontally, it would fall towards the Earth at the same rate the Earth would “fall” away

• This would result in the cannonball orbiting the Earth

Geostationary Satellite• A geostationary (or geosynchronous) satellite,

will always be above the same spot on Earth

• What is the orbital radius, altitude and speed of a geostationary satellite?– Use Newton’s Version of Kepler’s Law to solve

for orbital radius– Subtract Earth’s radius from orbital radius to

determine altitude– Set gravitational force equal to centripetal force

and solve for orbital speed

Weightlessness

Weightlessness

• The International Space Station orbits at an altitude of 226km; determine the force of gravity on an astronaut (65.kg) at this altitude and compare this to their weight on the surface of the Earth

• 594N at the ISS

• 638N on Earth

• Is the astronaut “weightless?”

Weightlessness• Weightlessness occurs

because objects are all falling towards the surface of the Earth at the same rate

• NASA simulates this on the “Vomit Comet” a high altitude aircraft that plunges toward Earth

Another way to look at “g”…

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Another way to look at gravitational potential energy of an object… (h is height but since it is arbitrary, it can be chosen as the distance from the center of the Earth to the position of the object…or r)

Some important orbital applications…

Geosynchronous means having an orbit around the Earth with a period of 24 hours

Gravitational Field• The strength of a gravitational

field can be determined using a test mass (mt)

• The mass should be very small compared to the mass creating the field

• A gravitational field will be measured in the units of N/kg

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Gravitational Field – One Mass• Arrows point toward

the mass showing the direction of force experienced by a mass at that point

• Circles represent equal potential (i.e. where the force experienced by a mass would be equivalent

Gravitational Field – Two Masses

• Note that earth’s field is much larger than the moon’s

• Small changes in the field as you approach the moon

Motion of a Particle in a Gravitational Field

• While the gravitational field lines show the direction of the force, they do not indicate the direction of motion

• Consider the behaviour of our solar system– http://janus.astro.umd.edu/javadir/orbits/ssv.ht

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Spacetime• One of Einstein’s

contributions is the concept of spacetime– In spacetime, space and time

are warped by massive objects

– The resulting curved spacetime results in objects following a curved path as they move through spacetime

– This is easily modeled by considering massive objects on a flexible sheet

Einstein viewed gravity and the motion of celestial objects, like planets, VERY differently…

(1875 – 1955)

Curved space-time effects both mass and light!

Black Holes

In 1916, Einstein published his theory of general relativity (GR), which discussed gravity and explained how the presence of matter causes space and time to be warped.

Light travel in curved spacetime

Photons of light passing near our Sun will move the same way through curved space.

They will “bend” around the Sun.

Time runs slower in curved spacetime

A fundamental tenet of GR is that time runs more slowly in curved spacetime.

The Principle of Equivalence, which states:

All local, freely falling, non-rotating laboratories are fully equivalent for the

performance of all physical experiments.

Gravitational Redshift•Another important effect of Einstein’s theory of GR is gravitational redshift photons lose energy as they try to escape from a strong gravitational field. •energy of a photon is inversely proportional to its wavelength. •Characteristic of systems containing high-density objects such as neutron stars and white dwarfs, although the effect would be particularly strong if the system contained a black hole.

The Schwarzschild radius, Rs, is given by

•Any star that collapses beyond its Schwarzschild radius is called a black hole. •coined by the American mathematical physicist John Wheeler in 1968.•A black hole is enclosed by an event horizon, the surface of which is described by a sphere with r = RS.•At the centre of the event horizon is a singularity, a point of zero volume and infinite density where all of the black hole’s mass is located.

Dark Matter