Transcript 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 Measurement
Time Distance
1
1
1
Total Distance
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
Free Fall Measurement
Time Distance
1
1
2
3
3
5
1
3
5
Total Distance
1
4
9
Free Fall Measurement
Time Distance
1
1
2
3
3
5
4
7
1
3
5
7
Total Distance
1
4
9
16
Free Fall Measurement
Time Distance
1
1
2
3
3
5
4
7
5
9
1
3
5
7
9
Total Distance
1
4
9
16
25
Free Fall Measurement
Time Distance
1
1
2
3
3
5
4
7
5
9
…
…
…
7
Total Distance
1
4
9
16
25
9
t2
1
3
5
Free-fall Velocity
v = vo + a t
Equations of Motion
v1= v0 + a t
x1= v0t + 1/2 a t2
velocity
distance
Initial Conditions: Start at rest, v0 = 0
Acceleration is gravity a = g ~ 10 m/s2
Free Fall Equations
v1= 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 Time
v1 = 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
in centimeters.
b) Multiply by 2
c) Divide by 980
d) Square Root
t=
x1
2 x1 980 cm/s2
Acceleration of Gravity
Things 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 = Dv/Dt
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
Path in the absence of g
1/2 g t2
1/2 g t2
Velocities
x = vox t
y = voy t - 1/2 g t2
Isaac Newton
Father 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 Law
Inertia
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 Law
F=ma
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=mg
Weight and Force
Our weight (W) is an example
of the force (F) we feel due to
the acceleration of gravity (g).
W = mg
(F = ma)
Mass No Matter
Lead 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 Down
Equating Newton's second law with gravity
F=ma
F = m GM/R2
m a = m GM/R2
m = apple,
m = human,
m = projectile,
m = moon?
What Goes Up, Must Come Down
Equating Newton's second law with gravity
F=ma
F = m GM/R2
m a = m GM/R2
a = GM/R2
Acceleration is GM/R2 ,
irregardless of the
mass m.
Surface Gravity
M
F = ma
F = mGM/R2
R
m
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
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/Rmoon2
• 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).
Cavendish
g = GM/R2
M = gR2/G
F = m GM/R2
GM = FR2/m
Mearth =
5.976 x 1024 kg
Gravity Works Everywhere
Earth and Moon
g = 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 p (60R)
t is the orbital period ~ 1 month
v is the moon's orbital speed = 2 p (60R)/(1 month)
Projectile Orbit Geometry
line 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 inch
s = d2/2r
s = [(2 p 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?
Orbits
Object 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.