Circular Motion: Circular Motion: GravitationGravitationChapter 5Chapter 5

5-1 Kinematics of 5-1 Kinematics of Uniform Circular MotionUniform Circular Motion

Uniform circular motion is when an object Uniform circular motion is when an object moves in a circle at constant speed.moves in a circle at constant speed.

ExamplesExamples The magnitude of the velocity remains The magnitude of the velocity remains

constant, but the direction continually constant, but the direction continually changes so the object accelerateschanges so the object accelerates

Acceleration is the change in velocity Acceleration is the change in velocity which includes both speed and directionwhich includes both speed and direction

Acceleration is defined Acceleration is defined as:as:

tv

tvva

12

Where Δv is the change in velocity during a short time interval Δt

5-1 Kinematics of Uniform Circular MotionLooking at the change in velocity in the limit that the time interval becomes infinitesimally small, we see that

(5-1)

Centripetal Centripetal AccelerationAcceleration

Centripetal means center-pointingCentripetal means center-pointing Or radial acceleration because it is Or radial acceleration because it is

directed along the radius.directed along the radius. aaRR

rl

vv

lrvv

Rearrange:

But, But, ΔΔl/l/ΔΔt is the linear speed, v, sot is the linear speed, v, sotl

rv

tvvaR

rvaR

2

SummarizeSummarize

An object moving in a circle of An object moving in a circle of radius r at a constant speed v radius r at a constant speed v has an acceleration ahas an acceleration aRR directed directed towards the center of the circle towards the center of the circle and the magnitude will be aand the magnitude will be aRR=v=v22/r/r

Acceleration is directed Acceleration is directed towards the center, but which towards the center, but which way is the velocity directed?way is the velocity directed?

In which direction would the In which direction would the object go if the string broke?object go if the string broke?

Tangential to the circle.Tangential to the circle.

Frequency and PeriodFrequency and Period Frequency is the number of revolutions per Frequency is the number of revolutions per

secondsecond Period is the time for one revolutionPeriod is the time for one revolution Remember one revolution would be equal to Remember one revolution would be equal to

the circumference of a circlethe circumference of a circle 22ππr so the speed around a circle can be r so the speed around a circle can be

written aswritten as

Trv 2

5-2 Dynamics of Uniform 5-2 Dynamics of Uniform Circular MotionCircular Motion

According to Newton’s second law According to Newton’s second law ΣΣF=maF=ma

This is true of linear motion as well as This is true of linear motion as well as circular motioncircular motion

SoSo ΣΣFFRR=ma=maRR= mv= mv22

r r So the net force must also be directed So the net force must also be directed

towards the centertowards the center

Centripetal force isn’t a new kind of force Centripetal force isn’t a new kind of force it just indicates the direction.it just indicates the direction.

This force must be applied by another This force must be applied by another object on the object in circular motion.object on the object in circular motion.

String on a ballString on a ball Earth’s gravitation pull on a satelliteEarth’s gravitation pull on a satellite

There is no outward force acting on the There is no outward force acting on the circular objects.circular objects.

Or centrifugal force.Or centrifugal force. It feels like it because when you swing a It feels like it because when you swing a

ball on a string you feel the “outward” ball on a string you feel the “outward” pull, but that is the equal and opposite pull, but that is the equal and opposite force the ball pulls on the string force the ball pulls on the string according to Newton’s second law.according to Newton’s second law.

If there was an outward force then when If there was an outward force then when the string broke the ball would fly outward the string broke the ball would fly outward and not linearly. and not linearly.

Vertical CircleVertical Circle

5-3 Highway Curves5-3 Highway Curves Banked and unbankedBanked and unbanked

5-5 Centrifugation5-5 Centrifugation Used to separate materialsUsed to separate materials Particles have a tendency to resist Particles have a tendency to resist

change in motion (Inertia) so they go change in motion (Inertia) so they go towards the bottom of test tubetowards the bottom of test tube

Causes rapid sedimentationCauses rapid sedimentation

5-6 Newton’s Law of 5-6 Newton’s Law of GravitationGravitation

Newton’s wondered about the forces that kept the Newton’s wondered about the forces that kept the circular orbit of the Moon around the Earthcircular orbit of the Moon around the Earth

Also knowing that falling object accelerate they Also knowing that falling object accelerate they must have a force acting on them alsomust have a force acting on them also

When an object has a force exerted When an object has a force exerted onon it, that force it, that force must be exerted must be exerted byby another object another object

Newton concluded that that other object is EarthNewton concluded that that other object is Earth This is even the case with the moonThis is even the case with the moon In fact, all object have a force of attraction between In fact, all object have a force of attraction between

themthem

Newton figured that Newton figured that the force decrease the force decrease as the distance as the distance between object between object increasedincreased

Where r is the Where r is the distance from center distance from center of object to center of of object to center of objectobject

2

1r

Fg

He also determined He also determined thatthat

objectEg mmF

ConclusionConclusion Every particle in the universe attracts every Every particle in the universe attracts every

other particle in the universe with a force other particle in the universe with a force that is proportional to the product of their that is proportional to the product of their masses and inversely proportional to the masses and inversely proportional to the square of the distances between them. square of the distances between them. This force acts along the line joining the This force acts along the line joining the two particles.two particles.

221

rmmGF

Where G=6.67 x 10-11 Nm2/kg2

5-7 Gravity Near the 5-7 Gravity Near the Earth’s SurfaceEarth’s Surface

5-9 Kepler’s Laws5-9 Kepler’s Laws Before Newton there was German Before Newton there was German

Johannes KeplerJohannes Kepler He worked out a detailed description of He worked out a detailed description of

the motion of the planets around the sun.the motion of the planets around the sun. Now known as Kepler’s LawsNow known as Kepler’s Laws

SummarySummary 11stst law - The path of each planet about law - The path of each planet about

the Sun is an ellipse with the Sun at one the Sun is an ellipse with the Sun at one focusfocus

22ndnd law – Each planet moves so that an law – Each planet moves so that an imaginary line drawn from the Sun to the imaginary line drawn from the Sun to the planet sweeps out equal area in equal planet sweeps out equal area in equal time periods.time periods.

33rdrd law – The ratio of the squares of the law – The ratio of the squares of the period of any two planets revolving about period of any two planets revolving about the Sun is equal to the ratio of the cubes the Sun is equal to the ratio of the cubes of their mean distances from the Sunof their mean distances from the Sun

Kepler analyzed data to arrive at his Kepler analyzed data to arrive at his resultsresults

50 years later Newton derived Kepler’s 50 years later Newton derived Kepler’s law mathematically from the Universal law mathematically from the Universal gravitation and the laws of motion.gravitation and the laws of motion.

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2

1

2

2

1

rr

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