Centripetal Force and Centripetal Force and GravityGravity

Chapter 5Chapter 5

How do the planets move?How do the planets move?

Newton developed mathematical Newton developed mathematical understanding of planets using:understanding of planets using: DynamicsDynamics AstronomyAstronomy

Overcame the idea of “Centrifugal” force – Overcame the idea of “Centrifugal” force – objects are throw outwardobjects are throw outward Items released from a circle move TANGENT Items released from a circle move TANGENT

to the curveto the curve

Centripetal ForceCentripetal Force

Center-seeking Center-seeking force exerted that allows force exerted that allows an object to move in a curved pathan object to move in a curved path Can comes fromCan comes from

• Pull of stringPull of string• GravityGravity• MagnetismMagnetism• FrictionFriction• Normal ForceNormal Force

Force acts towards the centerForce acts towards the center

Centripetal AccelerationCentripetal Acceleration

Centripetal force causes the object to Centripetal force causes the object to move in a curved linemove in a curved line

Acceleration caused byAcceleration caused by Increasing velocityIncreasing velocity Decreasing velocityDecreasing velocity Changing directionChanging direction

Centripetal accelerationCentripetal acceleration

Centripetal acceleration formulaCentripetal acceleration formula

aacc = = vv²²/r/r

aacc = centripetal acceleration (m/sm/s²²))

v = velocity (m/s)

r = radius (m)

Center-Seeking ForcesCenter-Seeking Forces

If a mass is accelerating it must have a If a mass is accelerating it must have a force acting on itforce acting on itCentripetal ForceCentripetal Force

FFcc = ma = macc = mv = mv²²/r/r This is the force that tugs a body off its This is the force that tugs a body off its

straight-line coursestraight-line course

Example #1: Strings and Flat Surfaces  

Suppose that a mass is tied to the end of a string and is being whirled in a circle along the top of a frictionless table as shown in the diagram below. 

                                                                                                                   

 

A freebody diagram of the forces on the mass would show

                                                                                         

The tension is the unbalanced central force: T = Fc = mac, it is supplying the centripetal force

necessary to keep the block moving in its circular path. 

Example #2: Conical Pendulums  

Our next example is also an object on the end of string but this time it is a conical pendulum. Notice, that its path also tracks out a horizontal circle in which gravity is always perpendicular to the object's path. 

                                                                                                                                                                   

A freebody diagram of the mass on the end of the pendulum would show the following forces.  

                                                                                                                                                                           

T cos θ is balanced by the object's weight, mg. It is T sin θ that is the unbalanced central force that is supplying the centripetal force necessary to keep the block moving in its circular path: T sin θ = Fc = mac.

Example #3: Flat Curves  

Many times, friction is the source of the centripetal force. Suppose in our initial example that a car is traveling through a curve along a flat, level road. A freebody diagram of this situation would look very much like that of the block on the end of a string, except that friction would replace tension. 

                                                                                                                                                               

 

Friction is the unbalanced central force that is supplying the centripetal force necessary to keep the car moving along its horizontal circular path: f = Fc = mac.

 Since f = μN and N = mg on this horizontal surface, most problems usually ask you to solve for the minimum coefficient of friction required to keep the car on the road.

Banked CurvesBanked Curves

““Bank” a turn so that normal force exerted Bank” a turn so that normal force exerted by the road provides the centripetal force by the road provides the centripetal force

To calculate the angle to bank at a set To calculate the angle to bank at a set speed:speed:

tan tan θθ = = vv²²/gr/gr As long as you aren’t going over the As long as you aren’t going over the

recommended velocity, you should never recommended velocity, you should never slip off a banked road (even if the surface slip off a banked road (even if the surface is wet)is wet)

Great NotesGreat Notes

http://spiff.rit.edu/classes/phys211/lectureshttp://spiff.rit.edu/classes/phys211/lectures/bank/bank_all.html/bank/bank_all.html

Example #4: Banked Curves   

If instead, the curve is banked then there is a critical speed at which the coefficient of friction can equal zero and the car still travel through the curve without slipping out of its circular path . 

                                                                                                 

 

A freebody diagram of the forces acting on the car would show weight and a normal. Since the car is not sliding down the bank of the incline, but is instead traveling across the incline, components of the normal are examined.   

                                                                                                                                                  

 

N sin θ is the unbalanced central force; that is, N sin θ = Fc = mac. This component of the normal is supplying the

centripetal force necessary to keep the car moving through the banked curve.

Circular MotionCircular Motion

GravityGravity Understand the math behind the forceUnderstand the math behind the force

Newtonian Newtonian • reliable and simplereliable and simple• fails on the “Grand” scale of the galaxyfails on the “Grand” scale of the galaxy

Einstein’s Theory of RelativityEinstein’s Theory of Relativity• Relates gravity to “fabric” of space and timeRelates gravity to “fabric” of space and time• Complex math – not needed for daily Complex math – not needed for daily

experienceexperience Today – still exploring Today – still exploring

• String theoryString theory• Dark EnergyDark Energy

Law of Universal GravitationLaw of Universal Gravitation

Gravity force is related to masses of two Gravity force is related to masses of two bodies and the distancebodies and the distance FFGG αα mM/r²mM/r² Center-to-Center attraction between all forms Center-to-Center attraction between all forms

of matterof matter

Evolution of the LawEvolution of the Law

Many scientists worked to developMany scientists worked to develop Copernicus and Galileo– Similar matter attractedCopernicus and Galileo– Similar matter attracted KeplerKepler

• Argued that two stones in space would attract to each other, Argued that two stones in space would attract to each other, proportional to their massproportional to their mass

• Noticed that force decreases with distance Noticed that force decreases with distance Bullialdus – Attraction was in a line dropping off Bullialdus – Attraction was in a line dropping off

inversely squaredinversely squared Newton – related centripetal acceleration to Newton – related centripetal acceleration to

gravitational accelerationgravitational acceleration

Gravitational ConstantGravitational Constant

By adding a constant the proportion can By adding a constant the proportion can be made into a equalitybe made into a equality

Universal Gravitational ConstantUniversal Gravitational Constant 6.672 x 10-6.672 x 10-¹¹ Nm²/kg²¹¹ Nm²/kg²

Measured by Cavendish Measured by Cavendish

But G is so small…But G is so small…

Only really noticed when one of the Only really noticed when one of the masses is REALLY BIGmasses is REALLY BIG

Unlimited rangeUnlimited range Purely attractive – not weakened by Purely attractive – not weakened by

repulsionrepulsion

Cool ConclusionsCool Conclusions

Cavendish wanted to find the density of Cavendish wanted to find the density of earth when he did his “G” experimentearth when he did his “G” experiment gg(surface)(surface) = GM/R = GM/R² (solve for M ² (solve for M D=M/V) D=M/V)

Newton (although he didn’t have Newton (although he didn’t have Cavendish’s experiment) made a guess at Cavendish’s experiment) made a guess at density to come up with “g” for earthdensity to come up with “g” for earth

Imperfect EarthImperfect Earth

Not a uniform sphereNot a uniform sphere Hills and valleysHills and valleys Bulge at the North (pear-Bulge at the North (pear-

shaped)shaped) The spin of earth “throws” The spin of earth “throws”

the center outthe center out Moon interferesMoon interferes

Gravity is not constant Gravity is not constant everywhereeverywhere

The Cosmic ForceThe Cosmic Force

Johannes KeplerJohannes Kepler Interesting family lifeInteresting family life ““Inherited” his life’s Inherited” his life’s

work from Tycho work from Tycho BraheBrahe

Took two decades Took two decades to formulate his to formulate his “Three Laws of “Three Laws of Planetary Motion”Planetary Motion”

Laws of Planetary MotionLaws of Planetary Motion

First LawFirst Law– The – The planets move in planets move in elliptical orbits elliptical orbits with the Sun at with the Sun at one focusone focus The orbits are The orbits are

NEARLY NEARLY circular, but an circular, but an oval makes a oval makes a differencedifference

Laws of Planetary MotionLaws of Planetary Motion

Second Law-Second Law- As As a planet orbits the a planet orbits the Sun it moves in Sun it moves in such a way that a such a way that a line drawn from line drawn from the Sun to the the Sun to the planet sweeps out planet sweeps out equal areas in equal areas in equal time equal time intervalsintervals

Second lawSecond law

The speed will be The speed will be greater when near greater when near the sunthe sun

As it moves away, As it moves away, gravity slows it gravity slows it downdown

Idea is used to Idea is used to “sling-shot” rockets “sling-shot” rockets and probes through and probes through spacespace

Laws of Planetary MotionLaws of Planetary Motion

Third lawThird law – The ratio of the average distance – The ratio of the average distance from the Sun cubed to the period squared is from the Sun cubed to the period squared is the same constant value for all planetsthe same constant value for all planets

rr³/T² = C³/T² = Cr – distance to Sunr – distance to SunT – time to travel around the SunT – time to travel around the SunC – Solar Constant*C – Solar Constant*

**Different constants for sun, earth, other planets or starsDifferent constants for sun, earth, other planets or stars

Third lawThird law

Satellite OrbitsSatellite Orbits

Projectiles – Sail in a parabola until it hits Projectiles – Sail in a parabola until it hits the earththe earth

Fire it faster – go farther Fire it faster – go farther Finally – the earth would “fall away” Finally – the earth would “fall away”

Different VelocitiesDifferent Velocities

Orbital speedOrbital speed

When centripetal force equals gravitational When centripetal force equals gravitational force – the object stays in orbitforce – the object stays in orbit

GmM/rGmM/r² = mv²² = mv²oo/r/r Simplified Simplified

vvoo = √GM/r = √GM/r Circular orbital speedCircular orbital speed

Varying OrbitalsVarying Orbitals

If the velocity is more or less than the If the velocity is more or less than the circular orbitalcircular orbital Circle – speed v = vCircle – speed v = voo Elliptical – speed v < vElliptical – speed v < voo Large elliptical – speed v > vLarge elliptical – speed v > voo and and < < √2v√2voo

Parabola – Parabola – vv = = √2v√2voo

Hyperbola - Hyperbola - vv > > √2v√2voo

Effectively WeightlessEffectively Weightless

When in free-fall, you have no weightWhen in free-fall, you have no weight If you stand on a scale in a free falling If you stand on a scale in a free falling

elevatorelevator The scale would drop to zeroThe scale would drop to zero No normal force pushing back-upNo normal force pushing back-up Only gravity is actingOnly gravity is acting

Vomit ComitVomit Comit

Gravitational FieldGravitational Field When an object experiences forces over a When an object experiences forces over a

continuous range of locationscontinuous range of locations Graviton – hypothetical massless carrier of Graviton – hypothetical massless carrier of

gravitational interactiongravitational interaction Gravity – elusive study in physicsGravity – elusive study in physics