AP Physics 1 Chapter 5Circular Motion, Newton’s Universal Law of

Gravity, and Kepler’s Laws

Centripetal force at work!

Hammer throw

Tangential Velocity and Centripetal Acceleration

Roadway banking

Sitges Terramar in Spain(60 degree bank)

Roadway Banking at an extreme

Speed/Velocity in a Circle

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Consider an object moving in a circle around a specific origin. The DISTANCE the object covers in ONE REVOLUTION is called the CIRCUMFERENCE. The TIME that it takes to cover this distance is called the PERIOD.

Speed is the MAGNITUDE of the velocity. And while the speed may be constant, the VELOCITY is NOT. Since velocity is a vector with BOTH magnitude AND direction, we see that the direction o the velocity is ALWAYS changing.

We call this velocity, TANGENTIAL velocity as its direction is draw TANGENT to the circle.

Centripetal Acceleration

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Suppose we had a circle with angle, between 2 radaii. You may recall:

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Centripetal means “center seeking” so that means that the acceleration points towards the CENTER of the circle

Drawing the Directions correctly

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So for an object traveling in a counter-clockwise path. The velocity would be drawn TANGENT to the circle and the acceleration would be drawn TOWARDS the CENTER.

To find the MAGNITUDES of each we have:

Circular Motion and N.S.L

Recall that according to Newton’s Second Law, the acceleration is directly proportional to the Force. If this is true: ForcelCentripetaF

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Since the acceleration and the force are directly related, the force must ALSO point towards the center. This is called CENTRIPETAL FORCE.

NOTE: The centripetal force is a NET FORCE. It could be represented by one or more forces. So NEVER draw it in an F.B.D.

Examples

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The blade of a windshield wiper moves through an angle of 90 degrees in 0.28seconds. The tip of the blade moves on the arc of a circle that has a radius of 0.76m. What is the magnitude of the centripetal acceleration of the tip of the blade?

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What is the minimum coefficient of static friction necessary to allow a penny to rotate along a 33 1/3 rpm record (diameter= 0.300 m), whenthe penny is placed at the outer edge of the record?

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Top view

Side view

ExamplesVenus rotates slowly about its axis,

the period being 243 days. The mass of Venus is 4.87 x 1024 kg. Determine the radius for a synchronous satellite in orbit around Venus. (assume circular orbit)

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Examples

The maximum tension that a 0.50 m string can tolerate is 14 N. A 0.25-kg ball attached to this string is being whirled in a vertical circle. What is the maximum speed the ball can have (a) the top of the circle, (b)at the bottom of the circle?

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Newton’s Law of GravitationWhat causes YOU to be pulled down? THE EARTH….or

more specifically…the EARTH’S MASS. Anything that has MASS has a gravitational pull towards it.

MmFgWhat the proportionality above is saying is that for there to be a FORCE DUE TO GRAVITY on something there must be at least 2 masses involved, where one is larger than the other.

N.L.o.G.As you move AWAY from the earth, your DISTANCE increases and your FORCE DUE TO GRAVITY decrease. This is a special INVERSE relationship called an Inverse-Square.

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The “r” stands for SEPARATION DISTANCE and is the distance between the CENTERS OF MASS of the 2 objects. We us the symbol “r” as it symbolizes the radius. Gravitation is closely related to circular motion as you will discover later.

N.L.o.G – Putting it all together

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Kepler’s Laws

Testing Models

Geocentric (or Ptolemaic) means the Earth is at the center and motionless.

Heliocentric (or Copernican) means the Sun is at the center and motionless.

Scholars wanted to differentiate models by comparing the predictions with precise observations.

This originated the modern scientific method.

Kepler’s Work

Tycho Brahe led a team which collected data on the position of the planets (1580-1600 with no telescopes).

Mathematician Johannes Kepler was hired by Brahe to analyze the data.

He took 20 years of data on position and relative distance.

No calculus, no graph paper, no log tables.

Both Ptolemy and Copernicus were wrong.

He determined 3 laws of planetary motion (1600-1630).

Kepler’s First Law

The orbit of a planet is an ellipse with the sun at one focus.

A path connecting the two foci to the ellipse always has the same length.

Orbital Description

An ellipse is described by two axes. Long – semimajor (a) Short – semiminor (b)

The area is ab (becomes r2 for a circle).

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Orbital Speed

The centripetal force is due to gravity. GMm/r2 = mv2/r v2 = GM/r

Larger radius orbit means slower speed.

Within an ellipse larger distance also gives slower speed.

Kepler’s Second Law

The line joining a planet and the sun sweeps equal areas in equal time.

The planet moves slowly here.

The planet moves quickly here.

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Orbital Period

The speed is related to the period in a circular orbit. v2 = GM/r (2r/T)2 = GM/r T2 = 42r3/GM

Larger radius orbit means longer period.

Within an ellipse, a larger semimajor axis also gives a longer period.

Kepler’s Third Law

The square of a planet’s period is proportional to the cube of the length of the orbit’s semimajor axis. T2/a3 = constant The constant is the same for all objects orbiting

the Sun

semimajor axis: a

direction of orbit

The time for one orbit is one period: T

Hyperbolic Orbits

Some satellites have so much speed that gravity can’t hold them in an orbit.

These objects take a hyperbolic orbit that never returns.