Centripetal Force and Gravity Chapter 5

How do the planets move?  Newton developed mathematical understanding of planets using: Dynamics Dynamics Astronomy Astronomy  Overcame the idea of “Centrifugal” force – objects are throw outward Items released from a circle move TANGENT to the curve Items released from a circle move TANGENT to the curve

Centripetal Force  Center-seeking force exerted that allows an object to move in a curved path Can comes from Can comes from Pull of stringPull of string GravityGravity MagnetismMagnetism FrictionFriction Normal ForceNormal Force Force acts towards the center Force acts towards the center

Centripetal Acceleration  Centripetal force causes the object to move in a curved line  Acceleration caused by Increasing velocity Increasing velocity Decreasing velocity Decreasing velocity Changing direction Changing direction

Centripetal acceleration  Centripetal acceleration formula a c = v²/r a c = v²/r a c m/s²) a c = centripetal acceleration ( m/s²) v = velocity (m/s) r = radius (m)

Center-Seeking Forces  If a mass is accelerating it must have a force acting on it Centripetal Force F c = ma c = mv²/r  This is the force that tugs a body off its straight-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 = F c = ma c, 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 = F c = ma c.

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 = F c = ma c. 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 Curves  “Bank” a turn so that normal force exerted by the road provides the centripetal force  To calculate the angle to bank at a set speed: tan θ = v²/gr  As long as you aren’t going over the recommended velocity, you should never slip off a banked road (even if the surface is wet)

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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 = F c = ma c. This component of the normal is supplying the centripetal force necessary to keep the car moving through the banked curve.

Circular Motion

Gravity  Understand 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 Relativity Einstein’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 experienceComplex math – not needed for daily experience Today – still exploring Today – still exploring String theoryString theory Dark EnergyDark Energy

Law of Universal Gravitation  Gravity force is related to masses of two bodies and the distance F G α mM/r² F G α mM/r² Center-to-Center attraction between all forms of matter Center-to-Center attraction between all forms of matter

Evolution of the Law  Many scientists worked to develop Copernicus and Galileo– Similar matter attracted Copernicus and Galileo– Similar matter attracted Kepler Kepler Argued that two stones in space would attract to each other, proportional to their mass Argued that two stones in space would attract to each other, proportional to their mass Noticed that force decreases with distanceNoticed that force decreases with distance Bullialdus – Attraction was in a line dropping off inversely squared Bullialdus – Attraction was in a line dropping off inversely squared Newton – related centripetal acceleration to gravitational acceleration Newton – related centripetal acceleration to gravitational acceleration

Gravitational Constant  By adding a constant the proportion can be made into a equality  Universal Gravitational Constant x 10-¹¹ Nm²/kg² x 10-¹¹ Nm²/kg²  Measured by Cavendish

But G is so small…  Only really noticed when one of the masses is REALLY BIG  Unlimited range  Purely attractive – not weakened by repulsion

Cool Conclusions  Cavendish wanted to find the density of earth when he did his “G” experiment g (surface) = GM/R² (solve for M  D=M/V) g (surface) = GM/R² (solve for M  D=M/V)  Newton (although he didn’t have Cavendish’s experiment) made a guess at density to come up with “g” for earth

Imperfect Earth  Not a uniform sphere Hills and valleys Hills and valleys Bulge at the North (pear- shaped) Bulge at the North (pear- shaped) The spin of earth “throws” the center out The spin of earth “throws” the center out Moon interferes Moon interferes  Gravity is not constant everywhere

The Cosmic Force  Johannes Kepler Interesting family life Interesting family life “Inherited” his life’s work from Tycho Brahe “Inherited” his life’s work from Tycho Brahe Took two decades to formulate his “Three Laws of Planetary Motion” Took two decades to formulate his “Three Laws of Planetary Motion”

Laws of Planetary Motion  First Law– The planets move in elliptical orbits with the Sun at one focus The orbits are NEARLY circular, but an oval makes a difference The orbits are NEARLY circular, but an oval makes a difference

Laws of Planetary Motion  Second Law- As a planet orbits the Sun it moves in such a way that a line drawn from the Sun to the planet sweeps out equal areas in equal time intervals

Second law  The speed will be greater when near the sun  As it moves away, gravity slows it down  Idea is used to “sling-shot” rockets and probes through space

Laws of Planetary Motion  Third law – The ratio of the average distance from the Sun cubed to the period squared is the same constant value for all planets r³/T² = C r – distance to Sun T – time to travel around the Sun C – Solar Constant* * Different constants for sun, earth, other planets or stars

Third law

Satellite Orbits  Projectiles – Sail in a parabola until it hits the earth  Fire it faster – go farther  Finally – the earth would “fall away”

Different Velocities

Orbital speed  When centripetal force equals gravitational force – the object stays in orbit  GmM/r² = mv² o /r  Simplified v o = √GM/r v o = √GM/r Circular orbital speed Circular orbital speed

Varying Orbitals  If the velocity is more or less than the circular orbital Circle – speed v = v o Circle – speed v = v o Elliptical – speed v < v o Elliptical – speed v < v o Large elliptical – speed v > v o and v o and < √2v o Parabola – v = √2v o Parabola – v = √2v o Hyperbola - v > √2v o Hyperbola - v > √2v o

Effectively Weightless  When in free-fall, you have no weight  If you stand on a scale in a free falling elevator The scale would drop to zero The scale would drop to zero No normal force pushing back-up No normal force pushing back-up Only gravity is acting Only gravity is acting

Vomit Comit

Gravitational Field  When an object experiences forces over a continuous range of locations  Graviton – hypothetical massless carrier of gravitational interaction  Gravity – elusive study in physics