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LESSON: Physics
NAME-SURNAME:
UNIT: Force and Motion SUBJECT: Universal Gravitation
CLASS:11
NUMBER:
In this chapter, we study the law of universal gravitation. We emphasize a description of
planetary motion because astronomical data provide an important test of this law’s validity.
We then show that the laws of planetary motion developed by Johannes Kepler follow from
the law of universal gravitation and the principle of conservation of angular momentum. We
conclude by deriving a general expression for the gravitational potential energy of a system
and examining the energetic of planetary and satellite motion.
Newton’s law of universal gravitation states that every particle in the Universe attracts
every other particle with a force that is directly proportional to the product of their masses and
inversely proportional to the square of the distance between them.
If the particles have masses m1 and m2 and are separated by a distance r, the magnitude of this
gravitational force is
where G is a constant, called the universal gravitational constant. Its value in SI units is
G =6.674 x 10-21 Nm2/kg2
This force is directed toward the center of the Earth.
Free-Fall Acceleration and the Gravitational Force
We have called the magnitude of the gravitational force on an object near the
Earth’s surface the weight of the object, where the weight is given by Equation
…………………
Equation ……………….is another expression for this force.
Therefore, we can set Equations equal to each other to obtain
Now consider an object of mass m located a distance h above the Earth’s surface or a distance
r from the Earth’s center, where r =RE + h. The magnitude of the gravitational force acting on
this object is
g decreases with increasing altitude. Because an object’s weight is mg, we see that as r →∞
the weight of the object approaches zero.
1) The gravitational force between two
objects is proportional to
A) the distance between the two objects.
B) the square of the distance between the
two objects.
C) the product of the two objects.
D) the square of the product of the two
objects.
5) As a rocket moves away from the
Earth's surface, the rocket's weight
A) increases.
B) decreases.
C) remains the same.
D) depends on how fast it is moving.
2) Two objects attract each other
gravitationally. If the distance between
their centers is cut in half, the gravitational
force
A) is cut to one fourth.
B) is cut in half.
C) doubles.
D) quadruples
6) Suppose a satellite were orbiting the
Earth just above the surface. What is its
centripetal acceleration?
A) smaller than g
B) equal to g
C) larger than g
D) Impossible to say without knowing the
mass.
3) Two objects, with masses m1 and m2,
are originally a distance r apart. The
magnitude of the gravitational force
between them is F. The masses are
changed to 2m1 and 2m2, and the distance
is changed to 4r. What is the magnitude of
the new gravitational force?
A) F/16
B) F/4
C) 16F
D) 4F
4) The acceleration of gravity on the
Moon is one-sixth what it is on Earth. An
object of mass 72 kg is taken to the Moon.
What is its mass there?
A) 12 kg
B) 72 kg
C) 72 N
D) 12 N
7) A hypothetical planet has a mass of half
that of the Earth and a radius of twice that
of the Earth. What is the acceleration due
to gravity on the planet in terms of g, the
acceleration due to gravity at the Earth?
A) g
B) g/2
C) g/4
D) g/8
8) The acceleration of gravity on the
Moon is one-sixth what it is on Earth. The
radius of the Moon is one-fourth that of the
Earth. What is the Moon's mass compared
to the Earth's?
A) 1/6
B) 1/16
C) 1/24
D) 1/96
9) The gravitational attractive force
between two masses is F. If the masses are
moved to half of their initial distance, what
is the gravitational attractive force?
A) 4F
B) 2F
C) F/2
D) F/4
10) An astronaut goes out for a "spacewalk" at a distance above the Earth equal
to the radius of the Earth. What is her
acceleration due to gravity?
A) zero
B) g
C) g/2
D) g/4
11) A satellite encircles Mars at a distance
above its surface equal to 3 times the
radius of Mars. The acceleration of gravity
of the satellite, as compared to the
acceleration of gravity on the surface of
Mars, is
A) zero.
B) the same.
C) one-third as much.
D) one-sixteenth as much.
12) An object weighs 432 N on the surface
of the Earth. The Earth has radius r. If the
object is raised to a height of 3r above the
Earth's surface, what is its weight?
A) 432 N
B) 48 N
C) 27 N
D) 0 N
Homework 1) During a lunar eclipse, the Moon, Earth, and Sun all lie on the same line, with
the Earth between the Moon and the Sun. The Moon has a mass of 7.36 × 1022 kg; the Earth
has a mass of 5.98 × 1024 kg; and the Sun has a mass of 1.99 × 1030 kg. The separation
between the Moon and the Earth is given by 3.84 × 108 m; the separation between the Earth
and the Sun is given by 1.496 × 1011 m.
(a) Calculate the force exerted on the Earth by the Moon.
(b) Calculate the force exerted on the Earth by the Sun.
(c) Calculate the net force exerted on the Earth by the Moon and the Sun.
Homework2) A 2.10-kg brass ball is transported to the Moon.
(a) Calculate the acceleration due to gravity on the Moon. The radius of the Moon is 1.74 ×
106 m and the mass of the Moon is 7.35 × 1022 kg.
(b) Determine the mass of the brass ball on the Earth and on the Moon.
(c) Determine the weight of the brass ball on the Earth.
(d) Determine the weight of the brass ball on the Moon.
Satellites and “Weightlessness”
Satellites are routinely put into orbit around the Earth. The
tangential speed must be high enough so that the satellite does
not return to Earth, but not so high that it escapes Earth’s gravity
altogether.
The satellite is kept in orbit by its speed – it is continually falling, but the
Earth curves from underneath it.
Consider a satellite of mass m moving in a circular orbit around the Earth at a constant speed
v and at an altitude h above the Earth’s surface as illustrated in Figure
(A) Determine the speed of satellite in terms of G, h, RE (the radius of the Earth),
and ME (the mass of the Earth).
SOLUTION
Conceptualize Imagine the satellite moving around the Earth in a circular orbit under the
influence of the gravitational force. This motion is similar to that of the space shuttle, the
Hubble Space Telescope, and other objects in orbit around the Earth.
Categorize The satellite must have a centripetal acceleration. Therefore, we categorize the
satellite as a particle under a net force and a particle in uniform circular motion.
Analyze The only external force acting on the satellite is the gravitational force, which acts
toward the center of the Earth and keeps the satellite in its circular orbit
Apply the particle under a net force and particle in uniform circular motion models to the
satellite:
Solve for v, noting that the distance r from the center of the Earth to the satellite is r = RE +h
13) Consider a small satellite moving in a
circular orbit (radius r) about a spherical
planet (mass M). Which expression gives
this satellite's orbital velocity?
A) v = GM/r
B) (GM/r)1/2
C) GM/r2
D) (GM/r2)1/2
14) Satellite A has twice the mass of
satellite B, and rotates in the same orbit.
Compare the two satellite's speeds.
A) The speed of B is twice the speed of A.
B) The speed of B is half the speed of A.
C) The speed of B is one-fourth the speed
of A.
D) The speed of B is equal to the speed of
A.
15) A person is standing on a scale in an
elevator accelerating downward. Compare
the reading on the scale to the person's true
weight.
A) greater than their true weight
B) equal to their true weight
C) less than their true weight
D) zero
16) A satellite is in a low circular orbit
about the Earth (i.e., it just skims the
surface of the Earth). What is the speed of
the satellite? (The mean radius of the Earth
is 6.38 × 106 m.)
A) 5.9 km/s
B) 6.9 km/s
C) 7.9 km/s
D) 8.9 km/s
17) A satellite is in a low circular orbit
about the Earth (i.e., it just skims the
surface of the Earth). How long does it
take to make one revolution around the
Earth? (The mean radius of the Earth is
6.38 × 106 m.)
A) 81 min
B) 85 min
C) 89 min
D) 93 min
Homework3) A satellite is in circular orbit
230 km above the surface of the Earth. It
is observed to have a period of 89 min.
What is the mass of the Earth? (The mean
radius of the Earth is 6.38 × 106 m.)
A) 5.0 × 1024 kg
B) 5.5 × 1024 kg
C) 6.0 × 1024 kg
D) 6.5 × 1024 kg
Objects in orbit are said to experience
weightlessness. They do have a
gravitational force acting on them,
though!
The satellite and all its contents are in
free fall, so there is no normal force. This
is what leads to the experience of
weightlessness.
Kepler’s Laws
Kepler’s complete analysis of planetary motion is summarized in three statements known as
Kepler’s laws:
1. All planets move in elliptical orbits with the Sun at one focus.
2. The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time
intervals.
3. The square of the orbital period of any planet is proportional to the cube of the semimajor
axis of the elliptical orbit.
Because the gravitational force provides the centripetal acceleration of the planet as it moves
in a circle, we model the planet as a particle under a net force and as a particle in uniform
circular motion and incorporate Newton’s law of universal gravitation,
18) Who was the first person to realize
that the planets move in elliptical paths
around the Sun?
A) Kepler
B) Brahe
C) Einstein
D) Copernicus
19) The speed of Halley's Comet, while
traveling in its elliptical orbit around the
Sun,
A) is constant.
B) increases as it nears the Sun.
C) decreases as it nears the Sun.
D) is zero at two points in the orbit.
20) Let the average orbital radius of a
planet be r. Let the orbital period be T.
What quantity is constant for all planets
orbiting the Sun?
A) T/R
B) T/R2
C) T2/R3
D) T3/R2
21) A planet is discovered to orbit around
a star in the galaxy Andromeda, with the
same orbital diameter as the Earth around
our Sun. If that star has 4 times the mass
of our Sun, what will the period of
revolution of that new planet be, compared
to the Earth's orbital period?
A) one-fourth as much
B) one-half as much
C) twice as much
D) four times as much
Homework 4) Two moons orbit a planet in
nearly circular orbits. Moon A has orbital
radius r, and moon B has orbital radius 4r.
Moon A takes 20 days to complete one
orbit. How long does it take moon B to
complete an orbit?
A) 20 days
B) 80 days
C) 160 days
D) 320 days
Gravitational Potential Energy
the gravitational force between two particles varies as 1/r 2, we expect that a more general
potential energy function—one that is valid without the restriction of having to be near the
Earth’s surface—will be different from U = mgy.
Gravitational potential energy of the Earth–particle system
When two particles are at rest and separated by a distance r, an
external agent has to supply energy at least equal to Gm1m2/r to
separate the particles to an infinite distance. It is therefore
convenient to think of the absolute value of the potential energy as
the binding energy of the system. If the external agent supplies
energy greater than the binding energy, the excess energy of the
system is in the form of kinetic energy of the particles when the
particles are at an infinite separation.
Escape Speed
Suppose an object of mass m is projected vertically upward from the Earth’s surface with an
initial speed vi as illustrated in Figure .We can use energy considerations to find the minimum
value of the initial speed needed to allow the object to move infinitely far away from the Earth
As the object is projected upward from the surface of the Earth, v = vi
and r = ri = RE. When the object reaches its maximum altitude, v = vf
= 0 and r =rf = rmax. Because the total energy of the isolated object–
Earth system is constant,
Traveling at this minimum speed, the object continues to move farther and farther away from
the Earth as its speed asymptotically approaches zero. Letting rmax →∞ ` and taking vi = vesc
gives
This expression for vesc is independent of the mass of the object. In
other words, a spacecraft has the same escape speed as a molecule.
Furthermore, the result is independent of the direction of the velocity
and ignores air resistance.
22) a) What is the escape velocity on a planet radius 500 km and whose surface gravity is 3
m/s2?
b) How high will a particle rise if it leaves the surface of the planet with a vertical velocity of
1000m/s?
23) A rocket ship takes off from the earth on a trip to Jupiter with a stopover on the moon.
Find the amount of energy required to escape from these two bodies relative to that required
escaping from Earth.
Energy Considerations in Planetary and Satellite Motion
Consider an object of mass m moving with a speed v in the
vicinity of a massive object of mass M, where M >> m. The
system might be a planet moving around the Sun, a satellite
in orbit around the Earth, or a comet making a one-time flyby
of the Sun. If we assume the object of mass M is at rest in an
inertial reference frame, the total mechanical energy E of the
two-object system when the objects are separated by a
distance r is the sum of the kinetic energy of the object of
mass m and the potential energy of the system, given by
Equation
Newton’s second law applied to the object of mass m gives
the escape speed of an object in orbit around the Earth
24) a)Does it take more energy to get a satellite up to 1500 km above the earth surface than to put it in orbit once
it is there?
b) What about 3000 km? c) What about 4500 km? (radius of the earth is 6000km)