The Beginning of Modern Astronomy. Isaac Newton (1642-1727) . did research on optics (the properties of light) invented calculus discovered the three laws of Motion discovered the law of universal gravitation. Motion Concepts. inertia : resistance to a change in motion.
where a is the
centripetal acceleration, v is the speed, and r is the radius of the circle.Circular Motion, Centripetal Acceleration, and Centripetal Force
Example 1The average distance from Earth to the Moon is 3.84×108 m, and the moon’s average orbital speed is 1022 m/s. Calculate its centripetal acceleration.
mNewton’s Law of Gravity
Any two particles in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the distance between them.
The minus sign reminds us that the force is attractive.
G = 6.673×10-11 Nm2/kg2
M and m are the masses of the two particles.
r is the distance between their centers.
A spherically symmetric object is one whose mass is distributed equally in all directions. The force on a particle outside an object with spherical symmetry is the same as if all of of the object’s mass were concentrated at its center. This allows Newton’s law of gravity to be used for things like planets, which are almost spherically symmetric.
is negligible, the only force acting on it is gravity.
Fnet = FG
M = mass of Earth m = mass of object
R = radius of earthRelation Between Weight and Mass
weight= the gravitational force on an object.
Fnet = ma
The acceleration due to gravity is directly proportional to the mass of the planet and inversely proportional to the square of the distance from the planet’s center. It is usually denoted by the symbol g.
W = weight = mg
Near earth’s surface, a = g = 9.8 m/s2.
When air resistance is negligible, the acceleration of a falling body does not depend on its weight.
Newton’s second law of motion and his law of gravity enable us to determine the masses of planets and stars.
The acceleration due to gravity at the surface of the Earth is 9.8 m/s2, and the radius of the Earth is 6380 km. What is the mass of Earth?
From the moon’s orbital speed and its distance from the center of Earth, Newton knew that the centripetal acceleration of the Moon is 0.0027 m/s/s; this is the experimental value of the moon’s acceleration. Since the (average) distance from Earth to the Moon is about 60 times the radius of Earth, Newton’s law of gravity predicts that the acceleration should be
An astronaut whose weight is 150 lb on Earth is launched to an
altitude of twice earth’s radius. What is his weight at that altitude?
The agreement between this theoretical value and the experimental value was an important confirmation of Newton’s law of gravity.
When the net force on a particle is always directed toward a fixed point, its angular momentum relative to that point does not change with time; i.e., its angular momentum is conserved.
The gravitational force of the Sun on a planet is always directed toward the Sun, so the angular momentum of the planet relative to the sun is conserved.
It can be shown that the conservation of the angular momentum of a planet is equivalent to the statement that the line from the Sun to the planet sweeps out equal areas in equal times, so Kepler’s second law of planetary motion is equivalent to the law of conservation of angular momentum applied to a planet in orbit around the Sun.
Energy is the ability to move an object while exerting a force on it.
The energy of an object due to its motion is called kinetic energy. It is defined by the equation
Potential Energyis the energy that a group of objects has because of their relative positions. There is no single formula for potential energy.
When you exert a force on an object and cause it to move, you put energy into it. The process of putting mechanical energy into an object is called doing work on the object.
Radiative Energyis the energy of the electric and magnetic fields in electromagnetic radiation.
m = the mass of the planet (or satellite).
M = the mass of the Sun (or planet).
G = 6.673×10-11 Nm2/kg2.
Solving for M
The law of conservation of energy can be used to prove Kepler’s third law of planetary motion and add some detail to it.
The sidereal period of the Moon is 27.32 days, and its average distance from Earth is 384,000 km. Calculate the mass of Earth.
Assume that the mass (m) of the Moon is negligible compared to that of Earth (M).
P = 27.32 days = 2.732×101×8.64×104 s = 2.360×106 s
a = 3.84×105 km = 3.84×105×103 m= 3.84×108 m
G = 6.673×10-11Nm2/kg2
M = 6.02×1024 kg
The orbital speed of a satellite in a circular orbit of radius r around a planet of mass M is
where G = 6.67310-11Nm2/kg2.
Calculate the orbital speed of a satellite in a circular orbit 150 km above the surface of Earth. Assume that the radius of earth is 6380 km and its mass is 5.98×1024 kg.
r = 150 km + 6380 km = 6530 km
r = 6.53×103 x 103 m
r = 6.53×106 m
M = 5.98×1024 kg
Vc = 28,500 km/hr = 17,500 mi./hr
If an object is at a distance r from the center of a planet of mass M, it can escape from the planet if its speed is at least equal to Ve where
Ve is called the escape velocity.
r = 6.38×106 m
M = 5.98×1024 kgEscape Velocity
Calculate the escape velocity from the surface of Earth.
The length of a day increases by about 0.0023 seconds per century, and the Moon moves farther from Earth by about 3.8 cm per year. Why?
900 million years ago, earth’s day was 18 hours long.