Understanding
Motion, Energy, and Gravity
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Understanding Motion, Energy, and Gravity
Going Towards a Grand Synthesis

Galileo Galilei
• Credited with setting the standard for studying nature through
reliance on observation and experimentation to test hypotheses
• The heavens had similar features to the Earth (contrast Aristotle)
• Galileo was the first to develop our modern ideas of motion –
made use of the inclined plane
– Demonstrated that all objects at the Earth’s surface fall at the same rate
regardless of mass
– Proposed the concept of inertia that was to overthrow Aristotle’s notion
that the natural motion of all earthly objects is to come to rest.

René Descartes
• Extended Galileo’s notion of inertia along the Earth’s surface to
that of straight line motion
• Proposed three laws of motion which would inspire Newton to
create the now classical Three Laws of Motion

Robert Hooke
• Suggested circular planetary motion required a central force
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Understanding Motion, Energy, and Gravity
Isaac Newton – A Summary

Newton – In a Nutshell
• The year Galileo died - 1642 - is the year Isaac
Newton was born.
• Accomplishments (many after he graduated from
Trinity College in Cambridge and returned home
to escape the plague that hit the city in 1666)
–
–
–
–
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Invent reflecting telescope
Co-invent calculus
Develop particle theory of optics
Discover laws of motion
Discover universal law of gravitation
• Lucasian Professor of Mathematics at
Cambridge University
• “If I have seen further, it is by standing on the
shoulders of giants”, Newton to Hooke 1675/76
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Understanding Motion, Energy, and Gravity
Newton’s First Two Laws of Motion
 Inertia
is the property of an object
whereby it tends to maintain whatever
velocity it has.
 Newton’s First Law (Law of Inertia):
Unless an object is acted upon by a
net, outside force, the object will
maintain a constant speed in a
straight line.
 Note: Speed and direction = velocity
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Understanding Motion, Energy, and Gravity
Newton’s First Two Laws of Motion
 An
object at rest has a speed of zero.
 Newton’s first law says that a net
force is needed to change the speed
and/or direction of an object’s motion.
 Acceleration is a measure of how
rapidly the speed or direction of
motion of an object is changing.
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Understanding Motion, Energy, and Gravity
Newton’s First Two Laws of Motion

More definitions:
• Mass (M or m): quantity of inertia
– An intrinsic property of an object
– Not volume or weight
– SI unit of measurement is a kilogram (kg)
• Momentum (p): mass times velocity
• Weight (W): gravitational force between some object and a
planetary body on which the object rests
– Be careful with this definition
– On the Earth: 1 kilogram has an equivalent weight of 2.2 lbs.
• Density: Mass divided by Volume
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Understanding Motion, Energy, and Gravity
Newton’s First Two Laws of Motion
 Newton’s
Second Law
• Acceleration is inversely proportional to the
mass being accelerated
• Acceleration = net force / mass
• Force = mass × acceleration, or F = ma
• When the net force is zero, there is no
acceleration.
• More precise way of stating the 2nd law: “When
a net force acts on an object, it produces a
change of momentum of in the direction in
which the net force acts”
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Understanding Motion, Energy, and Gravity
Newton’s Third Law
 Newton’s
Third Law
• When object X exerts a force on object Y, object
Y exerts an equal and opposite force back on X
• The Third Law is sometimes stated as “For every
action there is an opposite and equal reaction,”
but the first statement is more precise in terms of
physical forces
• This law does not “feel” right – be careful not to
confuse force with acceleration
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Understanding Motion, Energy, and Gravity
Motion in a Circle
 Going
in Circles
• Motion of an object in a circle at constant
speed (uniform circular motion) is an
example of acceleration by changing
direction
• Centripetal (“center-seeking”) force is the
force directed toward the center of the
curve along which the object is moving
• What happens if the centripetal force is
removed?
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Understanding Motion, Energy, and Gravity
Conservation Laws
 Physical
Quantities That Do Not Change
• Under certain conditions, certain physical
quantities will not change in time
• These unchanging quantities are said to be
conserved
• Three important conservation laws for
astronomy
– Conservation of (Linear) Momentum
– Conservation of Angular Momentum
– Conservation of Energy
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Understanding Motion, Energy, and Gravity
Conservation of Momentum

Momentum (Along a Line) and Conservation
• The momentum of an object with mass m and velocity
v is given as
p = mv
• The momentum of a system of objects is
P = p1 + p2 + … = m1v1 + m2v2 + …
• If the absence of external forces acting on the
system, P remains constant for all time - this is the
Conservation of Momentum
• Examples: Rockets and billiard balls
• For more than one direction, conservation of
momentum is applied in each direction separately
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Understanding Motion, Energy, and Gravity
Conservation of Angular Momentum
• Angular Momentum and Conservation
– Spinning objects and objects in orbit are said to possess
angular momentum
– In the absence of a “twisting force” or torque, a spinning
object will maintain its angular momentum - this is the
Conservation of Angular Momentum
– Orbital angular momentum
• The orbital angular momentum, J, of an object is the
product of that object’s mass m, speed of rotation v, and
distance from the center of rotation r:
J = mvr
• The conservation of J means that (in the absence of an
outside torque) as the distance to the spin axis
decreases (contraction), the speed increases
• This is what Kepler really observed as his 2nd Law of
Planetary Motions (the Law of Equal Areas)
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Understanding Motion, Energy, and Gravity
Conservation of Angular Momentum
• Angular Momentum and Conservation (continued)
– Rotational angular momentum
• An object (like the Earth) will continue to spin at the
same rate as long as there is no net torque on it
– Precession is the result of an external torque
(observed for the Earth)
• In a system of objects, the total angular momentum can
be conserved (no outside torque), but the objects may
transfer rotational angular energy between themselves
– The slowing of the Earth’s day is due to the transfer
of rotational angular momentum of the Earth to
orbital angular momentum of the Moon
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Understanding Motion, Energy, and Gravity
Conservation of Energy

Energy and Conservation
• Energy is basically what can make matter move
• Three major categories of energy
– Kinetic Energy: Energy of motion (= mv2/2)
– Radiative Energy: Energy associated with light
– Potential Energy: “Stored” energy
• Conservation of Energy states that in an isolated
system, although energy may change from one
form to another, the total amount of energy must
remain constant
• MKS unit for energy: Joule
– 4,184 joules are in one food calorie
– Typical adult eats 2500 calories = 10 million joules
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Understanding Motion, Energy, and Gravity
Conservation of Energy

Energy and Conservation
(continued)
• Three sub-categories of energy
– Thermal Energy
• The kinetic energy of many particles
• Temperature of randomly moving particles is the average
kinetic energy of the particles
– Scales: Fahrenheit, Celsius, Kelvin)
• Thermal energy of a collection of particles is basically the
product of the system’s temperature and density
– Gravitational Potential Energy
• Depends on an object’s mass and how far it can fall
• Typically converts into kinetic and thermal energy
– Mass-Energy (also a form of potential energy)
• Mass itself is a form of energy
• E = mc2
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Understanding Motion, Energy, and Gravity
The Law of Universal Gravitation
 The
Law of Gravity
• States that between any two objects there is an
attractive force, the magnitude of which is directly
proportional to the mass of each object and
inversely proportional to the square of the
distance between the centers of the objects
(inverse square law).
• In equation form:
F = Gm1m2/r 2
• where G is the gravitational constant, m1 and m 2
are the masses, and r is the distance between
their centers
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Understanding Motion, Energy, and Gravity
The Law of Universal Gravitation

Universal Gravity
• Gravity not only makes objects fall to Earth, but keeps
the Moon in orbit around the Earth and keeps the
planets in orbit around the Sun.
• Newton extended the laws on Earth to include the
heavens and therefore explain the planets’ motions
and why Kepler’s laws worked.
• Evidence in Favor of the Universal Claim
– The distance from the center of the Earth to the Moon is
about 60 times the distance from the center of the Earth
to its surface
– The centripetal acceleration of the Moon should be
(1/60²) or 1/3600 of the acceleration of gravity on Earth.
– Newton’s calculations showed this to be the case and
confirmed the validity of his theory of gravitation
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Understanding Motion, Energy, and Gravity
Kepler’s Laws and Conics

Immediate Consequences from the Universal Law of
gravitation and the Second Law of Motion
• Kepler’s three laws result
• Kepler’s third law is generalized (referred to as Newton’s
Version of Kepler’s Third Law):
P2 = ka3/(m1 + m2)
where k = 4p2/G
• Besides ellipses (bound orbits), objects may also travel
along parabolas and hyperbolas (unbound orbits)
– Ellipses, parabolas and hyperbolas are referred to as conic
sections
• Any two objects orbit each other around their common
center of mass – the “center” being proportionately closer
the more massive of the two objects.
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Understanding Motion, Energy, and Gravity
Examples of Newton’s Laws

Surface gravity and weight
• F = ma & F = GMm/r2 implies a = GM/r2
• With r = R, where R is the radius of the object of
mass M, a = g = GM/R2
• Note independence of g with respect to m
• For comparisons, be careful with M and R
• Weight = mg = GMm/R2
 Escape
Velocity
• Found from conservation of energy
• vesc=(2GM/R)1/2
• For comparisons, be careful with M and R
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Understanding Motion, Energy, and Gravity
Examples of Newton’s Laws

Weightlessness
• Weight is the force that counters gravity creating a zero
net force
• Weightlessness is the absence of the countering force
• People in orbit around the Earth feel weightless
because gravity is not countered by a surface
connected to the Earth

Changing Orbits
• Objects in orbit around each other do not
spontaneously change into other orbital configurations.
• The orbital energy of the system must change through:
– Gravitational encounters (encounters with a third object)
– Atmospheric drag (friction that diverts kinetic energy into
other forms)
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Understanding Motion, Energy, and Gravity
Tides and Gravity
 Tides - Basics
• The Moon and Sun pull on the Earth causing the water to
rise producing tides
• The Moon provides 2/3 of the pull, the Sun 1/3
• Earth’s rotation provides daily rise and fall of tides
• The Moon’s revolution about the Earth causes bi-weekly
rise and fall of tides
– Spring tide: Moon and Sun on same or opposite side of the
Earth
– Neap tide: Moon and Sun at perpendicular angles to the
Earth
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Understanding Motion, Energy, and Gravity
Tides and Gravity
 Tides - Details
• Due to inverse square law of gravity, the gravitational pull
on the Earth’s side closest to the Moon is stronger than
the opposite side.
• This tidal force stretches the entire Earth creating the two
tidal bulges.
• Rotation of Earth keeps the bulge, through tidal friction,
just ahead of the Earth-Moon line
• Consequences
– Earth’s rotation will slow
– Moon will move away from Earth
• Tidal bulge of Moon caused by Earth has resulted in the
synchronous rotation of the Moon
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