Gravity

Gravity• Read Your Textbook: Foundations of Astronomy

– Chapter 5

• Homework Problems– Review Questions: 3, 4, 5, 9, 10– Review Problems: 1, 3, 4– Web Inquiries:

Galileo• Galileo Discovered:

– Objects fall at the same rate, independent of their mass.– The increasing rate of speed is uniform.– The distance they fall each second follows the odd

numbers.– The total distance fallen is proportional to the time2.

Free Fall MeasurementTime Distance Total Distance

1 1 1

Let 1 unit of distance =

the distance the object falls during the first second.

This turns out to be 4.9 m ~ 5 m

The acceleration is uniform, g = 9.8 m/s/s ~ 10 m/s/s

1

Free Fall MeasurementTime Distance Total Distance

1 1 1

2 3 4

3 5 9

1

3

5

Free Fall MeasurementTime Distance Total Distance

1 1 1

2 3 4

3 5 9

4 7 16

1

3

5

7

Free Fall MeasurementTime Distance Total Distance

1 1 1

2 3 4

3 5 9

4 7 16

5 9 25

1

3

5

7

9

Free Fall MeasurementTime Distance Total Distance

1 1 1

2 3 4

3 5 9

4 7 16

5 9 25

…

…

… t2

1

3

5

7

9

Free-fall Velocityv = vo + a t

Equations of Motionv1= v0 + a t velocity

x1= v0t + 1/2 a t2 distance

Initial Conditions: Start at rest, v0 = 0

Acceleration is gravity a = g ~ 10 m/s2

Free Fall Equationsv1= v0 + a t

x1= v0t + 1/2 a t2

Initial Conditions: Start at rest, v0 = 0

Acceleration is gravity a = g ~ 10 m/s2

v1 = g t (This is how the velocity changes with time)

x1 = 1/2 g t2 (This is how total distance changes with time)

Reaction Timev1 = g t

x1 = 1/2 g t2

SOLVE FOR TIME

x1 = 1/2 g t2

2 x1 = g t2

2x1 g = t2

t = 2x1 g

Reaction Time Experiment

t = 2x1 g

a) Measure the distance

that the ruler falls x1

in centimeters.

b) Multiply by 2

c) Divide by 980

d) Square Root

t = 2 x1 980 cm/s2

Acceleration of GravityThings that fall, accelerate at

9.8 m/sec/sec near the

Earth's surface.

This means velocity of a falling

body increases by 9.8 m/sec with

each passing second.

Acceleration is the

change in velocity over the

change in time. a = v/t

Horizontal and Vertical Motion

Projectiles• Galileo’s Trajectories

x = vox t

y = voy t - 1/2 g t2

The horizontal distance (x)

is just due to the initial

velocity in the horizontal

direction (vox).

Or, how much kinetic

energy is imparted to the

object.

Projectiles x = vox t

y = voy t - 1/2 g t2

Trajectory Modified By Gravity x = vox t

y = voy t - 1/2 g t2

1/2 g t21/2 g t2

Path in the absence of g

Velocities x = vox t

y = voy t - 1/2 g t2

Isaac NewtonFather of Modern Physics

Newton -- Born Dec 25, 1642 or Jan 4, 1643

(Pope Gregory's 11 days)

Published Principia in 1687. Invented modern physics and

calculus.

This book had an immense effect on physics and astronomy.

Newton’s 1st LawInertia

An object at rest remains at rest

OR

if in motion, moves at constant velocity in a straight line

UNLESS

acted on by a net external force.

Inertia : the property of matter that resists motion.

Mass : the measure of inertia.

Mass is an innate characteristic of any chunk of matter.

Weight is actually the Force experienced by a Mass due to g.

Newton’s 2nd LawF = m a

An acceleration, a, is produced when a force of

magnitude F acts upon an object of mass m.

Twice the force will give twice the acceleration.

If you try to move twice the mass with a given amount of

force you'll only produce half the original acceleration.

Weight is the force felt by a mass due to acceleration.

W = m g

Weight and ForceOur weight (W) is an example

of the force (F) we feel due to

the acceleration of gravity (g).

W = mg (F = ma)

Mass No MatterLead and wood balls accelerate at the same rate when

dropped from Pisa’s leaning tower.

A hammer and feather fall at same rate in a vacuum.

Apollo 15 astronauts tested Galileo's hypothesis on the Moon

Astronaut David R. Scott, Apollo 15 commander, watches a

geological hammer and a

feather hit the lunar surface

simultaneously in a test of

Galileo's law of

motion concerning falling

bodies.

Newton’s 3rd Law

For every action, there is an equal and opposite reaction.

Objects do not just act, they interact.

I pull on the Earth, it pulls on me.

Newton's Law of Universal Gravitation

Fgravity = m GM/R2

This means that the force of gravity between any

two bodies in the universe is equal to a constant

(the Gravitational Constant, G = 6.67x10-11 N-m2/kg2)

times the product of the masses of the two bodies in

question (m and M),

divided by the square of the distance between their

centers (R).

Newton's Law of Universal Gravitation

Fgravity = m GM/R2

Double the mass, double the force.

Double the distance, reduce the force by 1/4.

Triple both mass and distance?

Newton's Law of Universal Gravitation

Fgravity = m GM/R2

Double the mass, double the force.

Double the distance, reduce the force by 1/4.

Triple both mass and distance?

Reduce the force by 1/3. 3X from M, (1/3)2 from R

What Goes Up, Must Come DownEquating Newton's second law with gravity

F = m a F = m GM/R2

m a = m GM/R2

m = apple,

m = human,

m = projectile,

m = moon?

What Goes Up, Must Come DownEquating Newton's second law with gravity

F = m a F = m GM/R2

m a = m GM/R2

a = GM/R2

Acceleration is GM/R2 ,

irregardless of the

mass m.

Surface GravityF = ma F = mGM/R2

a = GM/R2

• All free-falling bodies accelerate uniformly independent of their mass.

• Acceleration depends only on the mass of the attracting body and the distance from its center.

• Earth’s Surface Gravity a = g = G Mearth/Rearth2

• g = 9.8 m/s/s

m

MR

Gee, its “g”• g = 9.8 m/s2

• Surface Gravity

BUT, note that it is

dependent on r. Near

the surface r = Rearth

Want to lose weight?

Hike to the top of a

hill. Acceleration due

to gravity will be less,

therefore your weight

will be less.

Lunar Surface Gravity• Mmoon = 0.123 Mearth ~ 1/10

• Rmoon = 0.270 Rearth ~ 1/4

• gmoon = G (0.123 Mearth)/(0.270Rearth)2 ~ 1/6 gearth

Moon Gravity• Moon’s Surface Gravity a = gmoon = G Mmoon/Rmoon

2

• gmoon = 1.6 m/s/s

• Weight on the moon, W = mgmoon

• Since gmoon/gearth = 1/6, Wmoon/Wearth = 1/6

• You will weigh 1/6 as much, but your mass on the moon is the same as mass on the earth!

Measurement of Mearth

"g" is called our surface gravitational acceleration,

and g = 9.8 m/sec/sec .

The value g depends on G (a constant),

the mass of the earth (M) and

the radius of the earth (R).

Cavendishg = GM/R2

M = gR2/G

F = m GM/R2

GM = FR2/m

Mearth =

5.976 x 1024 kg

Gravity Works Everywhere

Earth and Moong = GM/R2

g = GM/(60R)2 acceleration of the moon due to earth

g is 1/3600th as great at the moon

than it is at the earth surface.

If the distance fallen in 1 second is 4.9 meters at

the surface of the earth, the distance fallen at the

distance of the moon is 4.9/3600 meters = 1/20 inch!

This is the theoretical prediction of

Newton’s Gravitational Theory.

1/20 inch?• How far does the moon actually fall in 1 second?

v = d/t

d is the circumference of its orbit = 2 (60R)

t is the orbital period ~ 1 month

v is the moon's orbital speed = 2 (60R)/(1 month)

Projectile Orbit Geometryline segment 0A = 0D = r = 60R

line segment AC = d = v/t

line segment CD = s

r2 + d2 = (s+r)2

r2 + d2 = s2 + r2 + 2rs

d2 = 2rs + s2

d2 = 2rs

s = d2/2r

1/20 inchs = d2/2r

s = [(2 60R/1 month) 1 second]2/2(60R)

60R = 1.513x1010 inches

1 month = 27.32 days (sidereal)

s ~ 1/20 inch

Newton’s Orbit Cannon• How much Energy is Required?

OrbitsObject must have

kinetic energy

greater than the

gravitational

potential energy

needed to escape

the earth. The

velocity associated

with this kinetic

energy is the

escape velocity.