Transcript Newton’s Law of Gravitation

Newton’s Law of
Gravitation
Applications of
Circular Motion
Bellwork 01/19/11
A 1.50-kg bucket of water is tied by a rope and
whirled in a circle with a radius of 1.00 m. At
the top of the circular loop, the speed of the
bucket is 4.00 m/s. Determine the
acceleration, the net force and the individual
force values when the bucket is at the top of
the circular loop.
m = 1.5 kg
a = _____m/s/s
Fnet = ______ N
Bellwork Solution
Fgrav = m • g = 14.7 N (g is 9.8 m/s/s)
a = v2 / R = (4 • Fgrav)/1 = 16 m/s/s
Fnet = m • a = 1.5 kg •16 m/s/s
= 24 N, down
Fnet = Fgrav + Ftens, so
Ftens = Fnet - Fgrav
Ftens = 24 N - 14.7 N = 9.3 N
**Challenge: What if the
bucket was at the bottom
of the circle?**
m = 1.5 kg
a = ____m/s/s
Fnet = _____ N
Review Centripetal Acceleration
• Work with your best partner using the rally
coach method to answer the questions on
your worksheet in the section called
“Centripetal Acceleration”
linear
angular
a = (vf - vo)/t
α = (ωf - ωo)/t
vf = vo + at
ωf = ωo + αt
s = ½(vf + vo)t
θ = ½(ωf + ωo)t
s = vot + ½at2
θ = ωot + ½αt2
vf2 = vo2 + 2as
ωf2 = ωo2 + 2αθ
Einstein’s Ideas About Gravity
Masses cause a distortion in space
The larger the mass, the larger it’s distortion
The Laws (Newton’s and Kepler’s) describe WHAT happens.
The Theory of Gravity is the attempt to explain WHY the
laws are true.
• Demo: Write three observations for each of the masses in
“space”
– Is the gravitational force influenced by the radius of the
object?
– Is gravitational force influenced by the mass of the
object?
– Is the gravitational force influenced by the diameter
between two objects?
•
•
•
•
Newton’s Law of Gravitation
• Newton’s Law of Gravitation: the mutual
gravitational attraction between any two point
masses is directly proportional to the product of
their masses and inversely proportional to the
square of the distance between them.
• The magnitude of the gravitational force is given by
• Where G = the universal gravitation constant, G =
6.67x10-11 Nm2/kg2 .
Applications of Newton’s Second Law
• a = F/m Thus the acceleration due to gravity by
a mass, M acting on another object m is equal to
• At Earth’s surface, where ME and RE are the mass
and radius of the earth
At an altitude, h above the Earth
Gravitational Potential Energy
• The gravitational potential energy between
two masses is given by
• Note: the denominator is r, not r2 as in
Newton’s law of gravitation.
• The minus sign derives from the choice of
zero reference point,
where U = 0, which is r = ∞
Let Equations Guide Your Thinking
• Work with your awesome partner using the
rally coach method to answer the remaining
questions on your worksheet .
Example One: Bohr’s Model of the Atom
The Bohr model of the hydrogen atom consists of
a proton of mass 1.67x 10-27 kg and an orbiting
electron of mass 9.11x10-31 kg. In one of its
orbits, the electron is 5.3 x10-11 m from the
proton and in another orbit it is 10.6 x10-11 m
from the proton.
A. What are the mutual attractive forces when the
electron is in each of these orbits?
B. If the electron jumps from the larger orbit to the
smaller one, what is the change in the potential
energy of the atom?
Solution to Example 1
Given: m1 = 1.67-27 x 10 kg, m2 = 9.11x10-31 kg
r1 = 5.3 x10-11 m , r = 10.6-11 x10 m
A)
r2 = 2r1, so F2 /F1 = 1/4
Solution to Example 1, cont.
B) The change in gravitational potential energy, DU = U1 – U2 .
Therefore, the atom loses potential energy.
**BONUS: Predict the color of light emitted by the electron as
it transitions from one orbital to the next. E = hn
Example 2
• Calculate the acceleration due to gravity at
the surface of the Moon, where
• RM = 1750 km = 1.750 x 106 m
• MM = 7.4 x 1022 kg
• How does the magnitude of ag on the moon
compare to the value of ag = g on Earth?
Solution to Example 2
Given: RM = 1750 km = 1.750 x 106 m
MM = 7.4 x 1022 kg
Independent Practice
• P. 288
#17, 19, 20, 33, 25, 26, 29, 34
• P. 295
#104, 105
• P. 250-254
#80, 81, 84, 85, 90
Kepler’s Laws and Earth’s
Satellites
Objective: State and explain Kepler’s
laws of planetary motion and describe
the orbits and motions of satellites
Warm-Up
• In The Little Prince, the Prince visits a small
asteroid called B612. If the asteroid B612
has a radius of only 20.0 m and a mass of
1.00 x 104 kg, what is the acceleration due
to gravity on asteroid B612?
• What is the escape speed needed to
A) Escape the gravitational pull of Asteroid
B612?
B) What would happen if you jumped up on
Asteroid B612?
Kepler’s Laws
1. The Law of Orbits: Planets move in elliptical
orbits, with the Sun at one of the focal points.
2. A line from the Sun to a planet sweeps out
equal areas in equal lengths of time.
3. The square of the orbital period of a planet is
directly proportional to the cube of the
average distance of the planet from the Sun
(see p. 240 for derivation of the equation
below).
Example 3: Calculate the mass of Jupiter
• The planet Jupiter is the larges in the solar
system, both in volume and mass. Jupiter
has 62 known moons, the four largest being
discovered by Galileo in 1610. These moons
are Io, Callisto, Ganymede, and Europa. If Io
is an average distance of 4.22 x 105 km
from Jupiter and has an orbital period of
1.77 days, compute the mass of Jupiter.
Solution to Example 3
Given
r = 4.22 x 105km = 4.22 x 108 m
T = 1.77 days (86400 s/day)=1.53 x 105s
Earth’s Satellites
• KEo +PEo = KE + PE
by the conservation of energy
In terms of gravitational potential energy
Where vesc is the escape velocity – the initial
velocity required to escape from the surface
of the Earth.
Staying In Orbit
• On Earth, the escape velocity is
approximately 11 km/s or 7 mi/s
• A tangential velocity less than the escape
velocity is required for a satellite to remain in
orbit around the Earth (depending on the
altitude of the satellite’s orbit, h).
Guided Practice
• With your team
– Calculate the mass of the Sun based on orbital
data of the Earth-Sun system (book’s cover or
Appendix III)
– solve problem #107 on page 255.
Independent Practice
• Chapter 7 ConcepTest with clickers
• P. 254-255; # 91, 92, 93, 95, 97, 104, 106
• TEST FRIDAY 01/29/10
• EXAMPLE 4: Oliver, whose mass is 65 kg, and
Olivia, whose mass is 45 kg, sit 2.0 m apart in
their physics classroom.
– A) What is the force of gravitational attraction
between Oliver and Olivia?
– B) Why don’t they drift toward each other?
• EXAMPLE 5: Temba is standing in the lunch line
6.38 x 106 m from the center of the earth.
Earth’s mass is 5.98 x 1024 kg.
– A) What is the acceleration due to gravity?
– B) When Temba eats lunch and increases his
mass, does this change the acceleration due
to gravity?
• EXAMPLE 6: Earth has a mass of 5.98 x 1024
kg and a radius of 6.38 x 106 km. What is
the escape speed of a rocket launched on
earth in m/s?
• EXAMPLE 7: Compare this with the escape
speed of a rocket launched from the moon if
the moon has a mass of 7.35 x 1022 kg and
a radius of 1.74 x 106 m.
Bellwork
• Calculate the total gravitational potential
energy for the system in the diagram below.