GravityGravity

Don’t Let It Get You Down!Don’t Let It Get You Down!

The Truth About GravityThe Truth About Gravity

Gravity is a phenomenon and results in Gravity is a phenomenon and results in a force which can accelerate objects a force which can accelerate objects with mass.with mass.

““g” = acceleration due to the force of g” = acceleration due to the force of gravitygravity

The Importance of Being GravityThe Importance of Being Gravity Gravity has basic properties: Gravity has basic properties:

– (1) It is long-ranged and can act over (1) It is long-ranged and can act over cosmological distancescosmological distances

– (2) It supplies an attractive force between any (2) It supplies an attractive force between any 2 pieces of matter in the Universe. 2 pieces of matter in the Universe.

Although extremely weak, Although extremely weak, – It wins over cosmological distances It wins over cosmological distances – Most important force for the understanding of Most important force for the understanding of

the large scale structure + evolution of the the large scale structure + evolution of the Universe.Universe.

So, let us deal with GRAVITYSo, let us deal with GRAVITY

We’ll need a bit of a history lesson:•Brahe•Kepler•Newton•Einstein

Pay close attention, gravity has many

implications!

1) The planets move about the sun in elliptical orbits with the Sun at one focus.

2) The radius vector joining a planetto the sun sweeps over equal areas in equal intervals of time.

3) The square of the time of one compete revolution of a planet about its orbit is proportional to the cube of the orbit's semi-major axis

Kepler’s “Laws” of Planetary Motion Kepler’s “Laws” of Planetary Motion

Isaac NewtonIsaac Newton

Born 1642, the year Galileo Born 1642, the year Galileo dieddied

Loner, tinkerer, paranoidLoner, tinkerer, paranoid 1665-1666 Plague was very 1665-1666 Plague was very

good for himgood for him Suffered mental breakdown Suffered mental breakdown

16751675 Math, Chemistry, Theology, Math, Chemistry, Theology,

ParliamentParliament Died 1727Died 1727 Has his picture on the British Has his picture on the British

pound notepound note

He put the physics and mathematics toKepler’s Laws!

Universal Law of GravitationUniversal Law of Gravitation

All objects in the Universe attract each All objects in the Universe attract each other with a force that varies directly as other with a force that varies directly as the product of their masses and the product of their masses and inversely as the square of their inversely as the square of their separation from each other.separation from each other.

F = G m mF = G m m

rrgravity 1 2

2

General RelativityGeneral Relativity

Einstein’s Theory of Gravity, published 1915

Principle of Equivalence: Accelerations are indistinguishable from gravitational fields. They are equivalent.

So, for example, when you are in freefall (like in an orbiting Shuttle), your downward acceleration is just enough to cancel the gravitational force.

Einstein’s View of GravityEinstein’s View of Gravity

Gravity is due to the curvature of space time.

Space time is curved by mass.

PredictionsPredictions

Light would be bent by gravity

Tested by Arthur Eddington during solar eclipse. Confirmed!

Emission of gravitational radiation by accelerating objects

Tested by observations of binary pulsars. Confirmed to 14 decimal places!

Applications of Newton’s 2nd Applications of Newton’s 2nd Law and Einstein’s GRLaw and Einstein’s GR

Projectile MotionProjectile Motion

PendulumsPendulums

Black HolesBlack Holes

Projectile MotionProjectile Motion

Projectile - any object given an initial velocity which subsequently follows a path determined by the gravitational force acting on it, and by the frictional resistance of the atmosphere

bullet shot from a gun, rocket after the fuel is exhausted,thrown or batted baseball

TrajectoryTrajectory

Trajectory - the path followed by a projectileTrajectory - the path followed by a projectile

Our AssumptionsOur Assumptions

We will consider only short length trajectories so that:

Gravitational force is considered constant in magnitude and direction Earth is an inertial system Air resistance is negligible

In other words, we will examine motion in a vacuum on a flat, non-rotating Earth. In physics, we call this creating an

First, the ForcesFirst, the Forces

Only force acting on the projectile is its weight…remember we are in the Ideal Case.

X-axis is horizontal; y-axis is vertical; origin is point where projectile starts its free flight

So x-component of the force on the projectile is zero and y-component of the force is the weight, mg.

AccelerationAcceleration

Netwon’s 2nd law then tells us that the x-component of the acceleration is zero and the y-component is -g.

In other words, trajectory is a combination of a horizontal motion with constant velocity and a vertical component with constant acceleration.

ResultResult

Under these conditions, projectiles travel in trajectories which are parabolas.

You can derive the equation for the parabolic motion from Newton’s Laws!

Let’s PlayLet’s Play

PendulumsPendulums

mg

T

mg sin

For small angles, sin

Simple Harmonic Motion

L

x

Period = 2 L/g

Measure period of oscillation and length of pendulum, determine g!

Black HolesBlack Holes

A huge great enormous thing, like — like nothing. A A huge great enormous thing, like — like nothing. A huge big — well, like a — I don’t know — like an huge big — well, like a — I don’t know — like an enormous big nothing …enormous big nothing …

Piglet describes the Heffalump,Piglet describes the Heffalump,

in in Winnie the PoohWinnie the Pooh by A.A. Milne by A.A. Milne

HistoryHistory

Based on Newton’s theory of gravityBased on Newton’s theory of gravity

Proposed independently by:Proposed independently by:

– 1783 Rev. John Mitchell1783 Rev. John Mitchell

– 1796 Pierre Simon Laplace1796 Pierre Simon Laplace

Structure of a BlackholeStructure of a Blackhole

event horizoncurvature

singularity

Artist’s ViewArtist’s View

HST ViewHST View

Calculation of Critical RadiusCalculation of Critical Radius

Calculation ContinuedCalculation Continued

Earth: normal sizeEarth: normal size

Earth: Normal Size

Earth: half sizeEarth: half size

Earth: Half Size

Earth: Quarter Size

Earth: Black Hole Rbh = 2GM/c2

R = 9 mm

Earth as Depression in Earth as Depression in SpacetimeSpacetime

Blackhole is Bottomless Blackhole is Bottomless Abyss in SpacetimeAbyss in Spacetime

Bending of LightBending of Light

D

D

R

photon

d

d is the approximate distance the photon falls over the diameter D of an object with mass M

From Newton, We Know...From Newton, We Know...

F = ma = GMm/R2

Solving for a,a = GM/R2

If you accelerate at rate a for time t, you move a distance d ofd = 1/2 at2

Recall that t = D/c. Putting everything together, we see:

d = 1/2 GMD2/(c2R2)

Solving the EquationsSolving the Equations

D

D

R

d

d = 1/2 GMD2/(c2R2)

but D=2R

and = d/D

RESULT: = GM/c2R

Using GR, we would get 4 GM/c2R!

Making the PredictionMaking the Prediction

When you plug in the values for G, c, and the mass and radius of the Sun, you predict that light should be bent by 8.5 x 10-6 radians.

This corresponds to 1.75 seconds of arc.

Eddington measured 1.75…probably.

Light Benders, Mind BendersLight Benders, Mind Benders

QuickTime™ and aVideo decompressor

are needed to see this picture.

QuickTime™ and aVideo decompressor

are needed to see this picture.

QuickTime™ and aVideo decompressor

are needed to see this picture.

QuickTime™ and aVideo decompressor

are needed to see this picture.

Mass and WeightMass and Weight

Weight is a force. It is the resultant gravitational force exerted on a body with mass m by all the other bodies on the Universe.

Near Earth’s surface, the gravitational force from our planet dominates all others, so according to Newton’s Laws, we can write:

Weight = Fg = G m Me / R2 = mg

where Me is the mass of the Earth and R is the radius of the Earth.

But also...But also...

Since weight is a force, it is measured in units of force, namely Newtons.

Remember what mass is measured in units of?

Remember This!Remember This!

Mass is a fundamental, universal property. You have the same amount of mass no matter where you are in the Universe. The only thing that can change your mass is velocity…according to General Relativity. As you approach the speed of light, you become infinitely heavy.

Weight is not fundamental; its value depends on what your circumstances are in the Universe.