An object is traveling in a circle at a constant speed. What can you conclude about the situation?

Warm-Up: November 4, 2014

Uniform Circular Motion and Gravitation

AP Physics 1Unit 3

Rotation Angle

The arc length, Δs, is the distance traveled along a circular path.

The radius of curvature, r, is the radius of the circular path.

The rotation angle, Δθ, is the ratio of arc length to the radius of curvature.

Definitions

For one complete revolution, the arc length equals the circumference of the circle, 2πr.

We can define a radian so that

We can convert between radians and degrees using the conversion

Radians

22

r

r

r

s

revolution 1 radians 2

360 radians 2

Angular velocity is the rate of change of an angle.

We use radians for θ, so the units for ω are radians per second.

Angular velocity, ω, is analogous to linear velocity, v.

Angular Velocity

t

Comparing v and ω

r

s

tt

sv

Linear velocity is proportional to the distance from the center of rotation.

Linear velocity is largest at a point on the rim of a spinning object.

We call this linear speed v of a point on the rim the tangential speed.

Tangential Speed

r

vrv or

The tangential speed of a tire is the same as the speed of the car.

A larger radius tire rotating at the same angular velocity will produce a greater linear speed for the car.

Consider a Car Tire

Consider a Car Tire

Calculate the angular velocity of a 0.300 m radius car tire when the car travels at 15.0 m/s.

Example 6.1

Calculate the angular velocity of a 1.20 m radius truck tire when the truck travels at 15.0 m/s.

You-Try 6.1

Read Chapter 7 of your textbook.

Assignment

A record is spinning on a record player at 45 revolutions per minute.

a) A fly lands 5.0 cm from the center of the record. What is the fly’s linear speed as it rests on the record?

b) The fly then moves to a position 10.0 cm from the center. What is the fly’s new linear speed?

Warm-Up: November 5, 2014

What is the magnitude and direction of the acceleration for uniform circular motion?

Essential Question

In uniform circular motion, the direction of velocity is always changing, so there is always an acceleration.

The acceleration is always towards the center of the circle.

We call this centripetal acceleration. Centripetal means “toward the center” or

“center seeking.”

Centripetal Acceleration

Change in Velocity

This is the same formula as linear acceleration.

Centripetal Acceleration

t

vac

Triangles ABC and PQR are similar.

Magnitude of ac

r

s

r

r

v

v

sr

vv

t

s

r

v

t

v

Magnitude of ac

rr

vac

22

What is the magnitude of the centripetal acceleration of a car following a curve of radius 500. m at a speed of 25.0 m/s?

Compare the acceleration with that due to gravity.

Example 6.2

What is the magnitude of the acceleration of a point 7.50 cm from the axis of an ultracentrifuge spinning at 7.50x104 rev/min.

Compare the acceleration with that due to gravity.

You-Try 6.2

Ladybug Rotation rotation_en.jar Ladybug 2D Motion

ladybug-motion-2d_en.jar

PhET Simulations

An ordinary workshop grindstone has a radius of 7.50 cm and rotates at 6500 rev/min.

a) Calculate the magnitude of the centripetal acceleration at its edge in meters per second squared and convert it to multiples of g.

b) What is the linear speed of a point on its edge?

Warm-Up: November 6, 2014

What is the force that keeps an object in uniform circular motion?

Essential Question

Any net force causing uniform circular motion is called a centripetal force.

Centripetal force may be one force (such as the force of gravity between a satellite and the Earth), or a combination of forces (such as gravity and tension for a ball attached to a rope swung in a vertical circle).

Centripetal Force

Uniform Circular Motion

22

net

mrr

vmF

maF

FFF

C

CC

C

Always directed towards the center of the circle (same direction as centripetal acceleration).

Centripetal Force

Calculate the centripetal force exerted on a 900. kg car that negotiates a 500. m radius curve at 25.0 m/s.

Assuming an unbanked curve, find the minimum static coefficient of friction between the tires and the road that keeps the car from slipping.

Example 6.4

In banked curves, the slope of the road helps you negotiate the curve.

The greater the angle, θ, the faster you can take the curve.

Banked Curves

For ideal banking, the net external force equals the horizontal centripetal force in the absence of friction.

Ideal Banking

Write an expression for the angle of ideal banking.

You-Try (with partner)

Ideal Banking, No Friction

rg

v21tan

Some curves of some race courses are very steeply banked. Calculate the speed at which a 100. m radius curve banked at 65.0° should be driven if the road is frictionless.

You-Try 6.5

Questions on the Reading?

A person driving a car makes a sharp right turn and feels that she is being pushed to the left.

Is there really a force pushing her to the left?◦ If so, what force?◦ If not, what is happening?

Warm-Up: November 10, 2014

Fictitious Forces Non-inertial Reference Frames Coriolis Effect

Today’s Topics

Unreal forces that arise from motion, that may seem real because the observer’s reference frame is accelerating or rotating.

Fictitious Forces

In an inertial reference frame, all forces are real, meaning they have an identifiable physical origin.

A non-inertial reference frame is accelerating or rotating, which leads to fictitious forces.

Reference Frames

What is the reference frame?

Imagine you are riding a merry-go-round. You are rotating with the merry-go-round,

which becomes your frame of reference. Since the reference frame is non-inertial, you

feel a fictitious force trying to throw you off. This outward fictitious force is called

centrifugal force. In Earth’s (inertial) frame of reference, there

is no force trying to throw you off. Really, you must hold on to go in a circle instead of a straight line.

Centrifugal Force

Merry-Go-Round

Rotate very rapidly

Particles with more inertia (mass) move to the bottom of the test tube

Centrifuges

What would happen if you were standing on a rotating merry-go-round and rolled a ball towards the edge, directly away from the center?

What would you see? What would an observer outside of the

merry-go-round see?

Think-Pair-Share

Rolling Ball on Merry-Go-Round

The ball follows a straight path relative to Earth.

The ball appears to curve to the right on the merry-go-round.

The apparent curve to the right is from a fictitious force, called the Coriolis force.◦ Also called the Coriolis effect.

Rolling Ball on Merry-Go-Round

A non-rotating reference frame placed at the center of the Sun is very nearly an inertial one. Why is it not exactly an inertial frame?

Warm-Up: November 12, 2014

Finishing Monday’s notes

Earth is not an inertial reference frame (even though we usually consider it to be).

Many events are affected by Earth’s rotation.

Most consequences of Earth’s rotation are analogous to the merry-go-round.

Any motion in Earth’s northern hemisphere experiences a Coriolis force to the right.◦ Just as with the merry-go-round.◦ In the southern hemisphere, the Coriolis force is

to the left.

Coriolis Force

Hurricanes and other low-pressure systems rotate counterclockwise in the northern hemisphere.

Wind circulation around high-pressure systems is clockwise in the northern hemisphere.

Coriolis Effect

Questions?

Read Chapter 7 (pages 219-253) Page 253 #3, 19, 23, 25, 35, 40-46, 60,

75-78, 93, 102, 103

Assignment

Newton’s Universal Law of Gravitation

Objects near the surface of Earth fall to the surface.

Satellites orbit planets. Planets orbit stars. Stars orbit the center of the galaxy. The Earth is nearly spherical.

Effects of Gravity

The weakest of the four basic forces◦ Strong nuclear, weak nuclear, electromagnetic

First precisely defined by Sir Isaac Newton◦ Proposed an exact mathematical form that

showed the motion of heavenly bodies are conic sections.

The force of gravity is always attractive. The magnitude of the force depends only on

the masses of the objects and the distance between them.

Gravity

Every particle in the universe attracts every other particle in the universe.

Newton’s Universal Law of Gravitation

2r

GmMF

Thought to be the same everywhere in the universe.

First accurately measured by Henry Cavendish in 1798.

Universal Gravitational Constant

2

211

kg

mN1067.6

G

Cavendish Experiment

Near Earth’s Surface

2r

GmMmg

2r

GMg

26

242

211

m1038.6

kg1098.5kg

mN1067.6

g

This equation was used to determine the mass of Earth

a) Find the acceleration due to Earth’s gravity at the distance of the Moon (3.84x108 m).

b) Given that it takes 27.3 days for the Moon to orbit Earth, calculate the centripetal acceleration needed to keep the Moon in its orbit. Assume a circular orbit around a fixed Earth.

c) Compare your answers for parts (a) and (b).

Warm-Up: November 13, 2014

kg 1035.7

kg 1098.522

Moon

24Earth

m

m

The Moon’s orbit is slightly elliptical. Earth is not stationary.

◦ The Earth and the Moon rotate around their center of mass

◦ The Earth and Moon are in orbit around the sun The Moon is slowly moving away from the

Earth (about 4 cm per year).

Why are they different?

Read Chapter 7 (pages 219-253) Page 253 #3, 19, 23, 25, 35, 40-46, 60,

75-78, 93, 102, 103

Assignment

Gravitons? Bending of space-time?

Nobody really knows. Gravity is still an active research topic.

What causes gravity?

Earth’s Motions

Ocean tides are mainly from Moon’s gravity◦ Sun’s effect is about half of the Moon’s effect

Tidal period is about 12 hours, 25.2 minutes

Tides

Not to scale

Kepler’s Laws

Kepler’s Laws describe orbits that have the following characteristics:

1. A small mass m orbits a much larger mass M. This allows us to approximate M as stationary, as if from an inertial reference frame placed on M.

2. The system is isolated from other masses. We neglect any small effects from other masses.

Kepler’s Laws

The orbit of each planet around the Sun is an ellipse with the Sun at one focus.

Valid for all orbits satisfying the two previously stated conditions.

Kepler’s First Law

Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.

Valid for all orbits satisfying the two previously stated conditions.

Can be derived from conservation of angular momentum. (We will see angular momentum in a later unit).

Kepler’s Second Law

The ratios of the squares of the period of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun.

Valid for all orbits satisfying the two previously stated conditions.

Kepler’s Third Law

32

31

22

21

r

r

T

T

Given that the Moon orbits Earth each 27.3 days and that it is an average distance of 3.84x108 m from the center of the Earth, calculate the period of an artificial satellite at an average altitude of 1500 km above Earth’s surface.

You-Try (Warm-Up) m 1038.6

kg 1098.56

Earth

24Earth

r

m

Homework Questions?

Derivation of Kepler’s Third Law

r

vmmaF c

2

net

r

vm

r

GmM 2

2

2vr

GM

22

T

r

r

GM

2

224

T

r

r

GM

32

2 4r

GMT

The International Astronomical Union (IAU) defined a planet in 2006 as a celestial body that:

1) is in orbit around the Sun;2) has sufficient mass to assume hydrostatic

equilibrium, and3) has cleared the neighborhood around its

orbit.

Planets (and dwarf planets)

Who Killed Pluto?

Read Chapter 7 (pages 219-253) Page 253 #3, 19, 23, 25, 35, 40-46, 60,

75-78, 93, 102, 103

Assignment

Questions?

Review

Warm-Up: November 17, 2014 A runner taking part in the 200. m dash must run around the end of a track that has a circular arc with a radius of curvature of 30.0 m. If he completes the 200. m dash in 23.2 s and runs at constant speed throughout the race, what is the magnitude of his centripetal acceleration as he runs the curved portion of the track?

Homework Questions?

A ball is rolling around a horizontal curved path, as shown above. Where does the ball go after it leaves the curved path?

Think-Pair-Share

Two friends are having a conversation. Anna says a satellite in orbit is in freefall because the satellite keeps falling toward Earth. Tom says a satellite in orbit is not in freefall because the acceleration due to gravity is not 9.8 m/s2.

With whom do you agree and why?

Think-Pair-Share

What is the ideal speed to take a 100. m radius curve banked at a 20.0° angle?

You-Try

Friday! Test is not yet written

Test

With the person sitting next to you, write at least one multiple choice test question.

The question must deal with at least one topic from this unit.

There may be one or more than one correct answer.

Incorrect answers should be good detractors.

Good questions will be included on Friday’s test.

Test Preparation

a) The Sun orbits the Milky Way galaxy once each 2.60x108 years, with a roughly circular orbit averaging 3.00x104 light years in radius. (A light year is the distance traveled by light in one year. The speed of light is on your formula sheet.) Calculate the centripetal acceleration of the Sun in its galactic orbit. Does your result support the contention that a nearly inertial frame of reference can be located at the sun?

b) Calculate the average speed of the Sun in its galactic orbit. Does the answer surprise you?

Warm-Up: November 18, 2014

Homework Questions?

Continue working with your partner from yesterday.

By the end of class, turn in the following:◦ 2 or more multiple choice questions with correct

answers indicated◦ 1 or more free response problem◦ Worked-out solution (not just a final answer) to the

free response problem(s) Excellent questions will be included in Friday’s

test (perhaps with different numbers). You may copy the problems to share with your

classmates (or take a picture).

November 18, 2014