Universal Law of Universal Law of GravityGravity

Newton’s Universal Law of Newton’s Universal Law of GravitationGravitation

Between every two objects there is an Between every two objects there is an attractive force, the magnitude of which is attractive force, the magnitude of which is directly proportional to the mass of each directly proportional to the mass of each object and inversely proportional to the object and inversely proportional to the square of the distance between the centers of square of the distance between the centers of the objects.the objects.

Universal Law of GravityUniversal Law of Gravity

Force of gravity has magnitude given Force of gravity has magnitude given byby

(Gravity Force) = (G) x(Gravity Force) = (G) x

ObjectA

ObjectB

( Mass of Object A ) x ( Mass of Object B)

( Distance ) x ( Distance )

DISTANCE

Force Force

Equal and opposite forces(Newton’s Third law)

Universal Gravity Universal Gravity Constant, GConstant, G

In the formula for gravity force, we haveIn the formula for gravity force, we have

G G = 0.0000000000667 N m= 0.0000000000667 N m22 / kg / kg22

= 6.67 x 10= 6.67 x 10–11–11 N m N m22 / kg / kg22

The formula and the constant are called The formula and the constant are called “universal” because, up to now, this “universal” because, up to now, this theory predicts gravity anywhere in the theory predicts gravity anywhere in the universe.universe.

Sample ProblemSample Problem

Here is an example of using the Here is an example of using the formulaformula

(Gravity Force) = (G) x(Gravity Force) = (G) x

( Mass of Object A ) x ( Mass of Object B)

( Distance ) x ( Distance )

DISTANCE = Earth’s Radius

ForceObject B (Earth)

Object A (1 kg mass)

Sample ProblemSample Problem

Find gravity force for a 1 kg mass on Find gravity force for a 1 kg mass on surface of Earth.surface of Earth.

(Force) = ((Force) = (6.67 x 106.67 x 10–11–11) x) x

Value comes out to 9.8 NewtonsValue comes out to 9.8 Newtons

( 1 ) x ( 6 x 1024 )

( 6.38 x 106 )2

Universal Gravity Constant, GEarth’s Radius

Earth’s Mass

Sample Problem (cont.)Sample Problem (cont.)

Find gravity acceleration on a 1 kg Find gravity acceleration on a 1 kg mass.mass.

Using Newton’s Second Law,Using Newton’s Second Law,

(Acceleration) = =(Acceleration) = =

Answer is 9.8 m/sAnswer is 9.8 m/s22. . We just We just confirmed our value of a!!!confirmed our value of a!!!

(Force)(Mass)

( 9.8 N )(1 kg )

Sample ProblemSample Problem

Are you attracted to the person sitting Are you attracted to the person sitting next to you? Calculate the gravitational next to you? Calculate the gravitational attraction between you (assume 70 kg) attraction between you (assume 70 kg) and the person next to you (assume 65 and the person next to you (assume 65 kg) if you are 1.2 m apart.kg) if you are 1.2 m apart.

givensgivens formulaformula

substitutionsubstitution

unknownunknown

FgFg

Answer = 2.11 x 10Answer = 2.11 x 10-7 -7 NN

0.000000211 N0.000000211 N

Very Small!!! But anything with Very Small!!! But anything with mass has an attractive force.mass has an attractive force.

Sample ProblemSample Problem

Determine the force of gravitational Determine the force of gravitational attraction between the earth (m = attraction between the earth (m = 5.98 x 105.98 x 102424 kg) and a 70-kg physics kg) and a 70-kg physics student if the student is standing at student if the student is standing at sea level, a distance of 6.38 x 10sea level, a distance of 6.38 x 1066 m m from earth's center.from earth's center.

Newton and the MoonNewton and the Moon

Newton realized that Newton realized that Earth’s gravity was Earth’s gravity was the centripetal the centripetal force that kept the force that kept the moon in orbit.moon in orbit.

Also discovered that Also discovered that gravity was gravity was weaker at that weaker at that great distance.great distance.

Gravityforce

Gravity & DistanceGravity & Distance

We don’t notice We don’t notice that gravity gets that gravity gets weaker as we weaker as we move away from move away from Earth because Earth because we rarely go we rarely go very far.very far.

Moon is 30 Earth

diameters away

Value of g Value of g (acceleration due to gravity; m/s(acceleration due to gravity; m/s22))

Using FUsing Fgravgrav = m·g = m·g we can derive: we can derive:

Value of gValue of g

Can now find g anywhereCan now find g anywhere

g = G•m/rg = G•m/r22

Where Where

G = 6.67 x 10G = 6.67 x 10-11-11 N m N m22/kg/kg22

m = mass of planet in kgm = mass of planet in kg

r = radius of planet in metersr = radius of planet in meters

Known as “Inverse Known as “Inverse Square Law”Square Law”

Gravity force Gravity force weakens with weakens with distance as distance as the inverse of the inverse of the square of the square of the distance.the distance.

Geometric Geometric property of property of area and area and distance.distance. outer circle is twice Earth’s radius

Earth Gravity

1/4 Earth Gravity

Sample ProblemSample Problem

Find your weight if you have a mass Find your weight if you have a mass of 60 kg and are 2.1 x 10of 60 kg and are 2.1 x 1055 m above m above the earth’s surface.the earth’s surface.

SolutionSolutionGivensGivens: :

r = 2.1 x 10r = 2.1 x 1055 m + 6.37 x 10 m + 6.37 x 1066 m = 6.58 x 10 m = 6.58 x 1066 m m

mmyouyou = 60 kg = 60 kg

mmearthearth = 5.98 x 10 = 5.98 x 102424 kg kg

UnknownUnknown: : EquationsEquations: :

FFgg F = ma F = ma for weight F for weight Fgg = mg = mg

Fg = FgFg = Fg = (60kg) = (60kg) (9.2 m/s(9.2 m/s22))

mg = mg = G mG m11 m m22 = 553 N = 553 N

rr22

g = g = G mG m

rr22

g = 9.2 m/sg = 9.2 m/s22

WeightlessnessWeightlessnessIn deep space, far away In deep space, far away

from all stars, from all stars, planets, etc. there is planets, etc. there is almost no gravity almost no gravity force.force.

In orbit near Earth, In orbit near Earth, gravity is still strong gravity is still strong (only 10% less than (only 10% less than on surface).on surface).

Why are Shuttle and Why are Shuttle and Space Station Space Station astronauts astronauts ““weightless”?weightless”?

Earth is nearby

NASA’s “Vomit Comet”NASA’s “Vomit Comet”

NASA has a special NASA has a special airplane for airplane for training astronauts training astronauts in free-fall in free-fall weightless weightless conditions.conditions.

The “Vomit Comet” The “Vomit Comet” nickname tells you nickname tells you it’s quite a wild it’s quite a wild roller-coaster ride.roller-coaster ride. The plane flies between 20,000 and

30,000 feet, same as commercial flights.

Pow

erClim

b

WeightlessFreefall

Pull o

ut

of D

ive

Flight of the “Vomit Comet”

At the top of the arc, the plane’s trajectory is projectile motion.

Boeing 707 (modified)

OrbitsOrbits

Geosynchronous Orbit – orbits above Geosynchronous Orbit – orbits above the same point on the equator of the the same point on the equator of the earth at all timesearth at all times

GPS, cell phones, etc.GPS, cell phones, etc.

Orbits and Centripetal Orbits and Centripetal ForceForce

Gravity provides the centripetal force Gravity provides the centripetal force required for a satellite to move in a circle. required for a satellite to move in a circle.

FFgg = F = Fcc

mg = mvmg = mv22/r/r

g = vg = v22/r/r

Sample ProblemSample Problem

Calculate the speed needed for one Calculate the speed needed for one of the Direct TV satellites to orbit at of the Direct TV satellites to orbit at an altitude of 320,000 m.an altitude of 320,000 m.

Getting into OrbitGetting into OrbitRocket needs to lift aboveRocket needs to lift abovethe atmosphere and thenthe atmosphere and thenfire thrusters to acquire fire thrusters to acquire

thetherequired orbital speed ofrequired orbital speed ofabout 8 kilometers perabout 8 kilometers persecond.second.

Returning to Earth, Returning to Earth, air resistance slows theair resistance slows thespacecraft during reentry.spacecraft during reentry.

Elliptical OrbitsElliptical Orbits

For speeds higher than 8 km/s, the For speeds higher than 8 km/s, the orbit is elliptical instead of circular.orbit is elliptical instead of circular.

Escape SpeedEscape Speed

If speed exceeds 11.2 km/s then object escapes Earth because gravity weakens (as object gets further away) and never slows the object enough to return it back towards Earth.

Circular

Elliptical

Hyperbolic

Using the Law of Universal Using the Law of Universal GravitationGravitation

Newton’s Law is used and applied to Newton’s Law is used and applied to the motion of the planets about the the motion of the planets about the sun.sun.

Newton derived an equation from Newton derived an equation from the Law of Universal Gravitation to the Law of Universal Gravitation to describe the describe the Period of Planetary Period of Planetary Motion:Motion:

T² = (4T² = (4ππ²/Gm²/Gmss)r³)r³