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## Rotational Motion and The Law of Gravity

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Title: Rotational Motion and The Law of Gravity

1
Rotational Motion and The Law of Gravity
• Ch 7

2
Rotation and Revolution
• Two types of circular motion are rotation and
revolution.
• An axis is the straight line around which
rotation takes place.
• When an object turns about an internal axisthat
is, an axis located within the body of the
objectthe motion is called rotation, or spin.
• When an object turns about an external axis, the
motion is called revolution.

3
• The Ferris wheel turns about an axis.
• The Ferris wheel rotates, while the riders
• Earth undergoes both types of rotational motion.
• It revolves around the sun once every 365 ¼ days.
• It rotates around an axis passing through its
geographical poles once every 24 hours.

4
Rotational Motion
• Solid objects undergo rotational motion
• A point on a rotating object undergoes circular
motion
• Circular Motion is described in terms of the
angle through which the point on the object moves

5
Centripetal Acceleration
• Tangential speed (vt) depends on distance
• When tangential speed is constant, motion is
described as uniform circular motion

6
• An object moving in a circle at a constant speed
still has an acceleration due to its change in
direction
• Velocity is a vector so acceleration can be
produced by a change in magnitude and direction
• Centripetal Acceleration is acceleration caused
by a change in direction, directed toward the
center of a circular path
• ac Vt2 / r

7
Centripetal Acceleration
8
at and ac
• Are perpendicular and not the same thing
• at is due to changing speed
• ac is due to change in direction
• To find the total (at ac) use Pythagorean
theorem
• Direction of total acceleration can be found
using trig functions

9
Board Work
• A test car moves at a constant speed around a
circular track. If the car is 48.2 m from the
tracks center and has a centripetal acceleration
of 8.05 m/s2, what is the cars tangential speed?
• The tub of a washing machine has a radius of 34
cm. During the spin cycle, the wall of tub
rotates with a tangential speed of 5.5 m/s.
Calculate the centripetal acceleration of the
clothes against the tub

10
Causes of Circular Motion
• Centripetal Force force that maintains circular
motion
• This force is necessary for circular motion
• Ball moving in a circle ?v due to ? in direction
• ac is inward ac vt2/r
• Fc is used to change an objects straight line
inertia
• Fc mac mvt2 / r

11
Without a centripetal force, an object in motion
continues along a straight-line path.
With a centripetal force, an object in motion
will be accelerated and change its direction.
12
Ff and Fc
• Inertia is often misinterpreted as a force
• Fc is the force directed toward the center and is
necessary for circular motion
• Many times Fc is the force provided by friction
• If the force is lost the object leaves at a
tangent to the circular motion

13
Force that maintains circular motion
• A pilot is flying a small plane at 56.6 m/s in a
circular path with a radius of 188.5 m. If a
force of 18,900 N is needed to maintain the
pilots circular motion, what is the planes
mass?
• A 2000. kg car rounds a circular turn of radius
20.0 m If the road is flat and the coefficient
of static friction between the tires and the road
is 0.70, how fast can the car go without skidding?

14
Gravitational Force
• Orbiting objects are in free fall
• When objects are orbiting, the gravitational
force between the object and Earth is a
centripetal force that keeps the object in orbit

Each successive cannonball has a greater
initial speed, so the horizontal distance that
the ball travels increases. If the initial speed
is great enough, the curvature of Earth will
cause the cannonball to continue falling without
ever landing.
15
Newtons Hypothesis
• Newton compared motion of the moon to a
cannonball fired from the top of a high mountain.
• If a cannonball were fired with a small
horizontal speed, it would follow a parabolic
path and soon hit Earth below.
• Fired faster, its path would be less curved and
it would hit Earth farther away.
• If the cannonball were fired fast enough, its
path would become a circle and the cannonball
would circle indefinitely.

16
• This original drawing by Isaac Newton shows how a
projectile fired fast enough would fall around
Earth and become an Earth satellite.

17
• Both the orbiting cannonball and the moon have a
component of velocity parallel to Earths
surface.
• This sideways or tangential velocity is
sufficient to ensure nearly circular motion
around Earth rather than into it.
• With no resistance to reduce its speed, the moon
will continue falling around and around Earth
indefinitely.

18
The gravitational force attracts Earth and the
moon to each other. According to Newtons 3rd
Law.
19
Newtons Law of Gravitation
• Gravitational force the mutual force of
attraction between the particles of matter
• Keeps the planets orbiting around the sun
• Exists between any two masses regardless of size
or composition
• Fg is inversely proportional to distance
• Distance increases, gravity decreases

20
Newtons Law of Gravitation
• Fg is localized to the center of a spherical mass
• Fg G (m1m2/r2)
• G is gravitational constant 6.673e -11 Nm2/kg2

21
Newtons Law of Gravitation
• Find the Fg exerted on the moon (m7.36e22 kg) by
Earth (m5.98e24 kg) when the distance between
them is 3.84e8 m
• Find the distance between a 0.30 kg ball and a
0.40 kg ball if the magnitude of the Fg is
8.92e-11 N

22
Newtons Law of Universal Gravitation
• The value of G tells us that gravity is a very
weak force.
• It is the weakest of the presently known four
fundamental forces.
• We sense gravitation only when masses like that
of Earth are involved.

23
• Cavendishs first measure of G was called the
Weighing the Earth experiment.
• Once the value of G was known, the mass of Earth
was easily calculated.
• The force that Earth exerts on a mass of 1
kilogram at its surface is 10 newtons.
• The distance between the 1-kilogram mass and the
center of mass of Earth is Earths radius, 6.4
106 meters.
• from which the mass of Earth m1 6 1024
kilograms.

24
• When G was first measured in the 1700s,
newspapers everywhere announced the discovery as
one that measured the mass of Earth

25
Gravitational Field
• We can regard the moon as in contact with the
gravitational field of Earth.
• A gravitational field occupies the space
surrounding a massive body.
• A gravitational field is an example of a force
field, for any mass in the field space
experiences a force.
• Gravitational field strength equals free-fall
acceleration

26
• Field lines can also represent the pattern of
Earths gravitational field.
• The field lines are closer together where the
gravitational field is stronger.
• Any mass in the vicinity of Earth will be
accelerated in the direction of the field lines
at that location.
• Earths gravitational field follows the
inverse-square law.
• Earths gravitational field is strongest near
Earths surface and weaker at greater distances
from Earth

27
• Field lines represent the gravitational field

28
Applications of Gravity
• Weight changes with location
• Gravitational mass equals inertial mass
• On the surface of any planet, the value of g, as
well as your weight, will depend on the planets
• Your weight is less at the top of a mountain
because you are farther from the center of Earth.

29
• Newtons law of gravitation accounts for ocean
tides.
• High and low tides are partly due to the
gravitational force exerted on Earth by its moon.
• The tides result from the difference between the
gravitational force at Earths surface and at
Earths center.
• The moons attraction is stronger on Earths
oceans closer to the moon, and weaker on the
oceans farther from the moon.
• This is simply because the gravitational force is
weaker with increased distance.

30
• The two tidal bulges remain relatively fixed with
respect to the moon while Earth spins daily
beneath them.

31
• Earths tilt causes the two daily high tides to
be unequal.

32
Keplers Laws of Planetary Motion
• Newtons law of gravitation was preceded by
Keplers laws of planetary motion.
• Keplers laws of planetary motion are three
by the German astronomer Johannes Kepler.

33
• Kepler started as an assistant to Danish
astronomer Tycho Brahe, who headed the worlds
first great observatory in Denmark, prior to the
telescope.
• Using instruments called quadrants, Brahe
measured the positions of planets so accurately
that his measurements are still valid today.
• After Brahes death, Kepler devoted many years of
his life to the analysis of Brahes measurements.

34
• Keplers laws were developed a generation before
Newtons law of universal gravitation.
• Newton demonstrated that Keplers laws are
consistent with the law of universal gravitation.
• The fact that Keplers laws closely matched
observations gave additional support for Newtons
theory of gravitation.

35
• Keplers laws describe the motion of the planets.
• First Law Each planet travels in an elliptical
orbit around the sun, and the sun is at one of
the focal points.
• Second Law An imaginary line drawn from the sun
to any planet sweeps out equal areas in equal
time intervals.
• Third Law The square of a planets orbital
period (T 2) is proportional to the cube of the
average distance (r 3) between the planet and the
sun.

36
Keplers 1st Law
• Keplers expectation that the planets would move
in perfect circles around the sun was shattered
after years of effort.
• He found the paths to be ellipses.

37
Keplers 2nd Law
• According to Keplers second law, if the time a
planet takes to travel the arc on the left (?t1)
is equal to the time the planet takes to cover
the arc on the right (?t2), then the area A1 is
equal to the area A2.
• Thus, the planet travels faster when it is closer
to the sun and slower when it is farther away

38
• After ten years of searching for a connection
between the time it takes a planet to orbit the
sun and its distance from the sun, Kepler
discovered a third law.
• Kepler found that the square of any planets
period (T) is directly proportional to the cube
of its average orbital radius (r).
• Keplers third law states that T 2 ? r 3.
• The constant of proportionality is 4p 2/Gm, where
m is the mass of the object being orbited.

39
Board Work
• Magellan was the first planetary spacecraft to be
launched from a space shuttle. During the
spacecrafts fifth orbit around Venus, Magellan
traveled at a mean altitude of 361km. If the
orbit had been circular, what would Magellans
period and speed have been?

40
• Kepler was the first to coin the word satellite.
• He had no clear idea why the planets moved as he
discovered. He lacked a conceptual model.
• Kepler was familiar with Galileos concepts of
inertia and accelerated motion, but he failed to
apply them to his own work.
• Like Aristotle, he thought that the force on a
moving body would be in the same direction as the
bodys motion.
• Kepler never appreciated the concept of inertia.
Galileo, on the other hand, never appreciated
Keplers work and held to his conviction that the
planets move in circles.

41
Rotational Motion
• Rotational and translational motion can be
analyzed separately.
• For example, when a bowling ball strikes the
pins, the pins may spin in the air as they fly
backward.
• The pins have both rotational and translational
motion.
• Measure the ability of a force to rotate an
object.

42
Torque
• Torque is a quantity that measures the ability of
a force to rotate an object around some axis.
• How easily an object rotates on both how much
force is applied and on where the force is
applied.
• The perpendicular distance from the axis of
rotation to a line drawn along the direction of
the force is equal to d sin q and is called the
lever arm.
• t Fd sin q
• torque force ? lever arm

43
• The applied force may act at an angle.
• However, the direction of the lever arm (d sin q)
is always perpendicular to the direction of the
applied force, as shown here.

44
In each example, the cat is pushing on the door
at the same distance from the axis. To produce
the same torque, the cat must apply greater force
for smaller angles.
45
• Sign of Torque
• Torque is a vector quantity. In this textbook, we
will assign each torque a positive or negative
sign, depending on the direction the force tends
to rotate an object.
• We will use the convention that the sign of the
torque is positive if the rotation is
counterclockwise and negative if the rotation is
clockwise

46
Board Work
• A basketball is being pushed by two players
during tip-off. One player exerts an upward force
of 15 N at a perpendicular distance of 14 cm from
the axis of rotation. The second player applies a
downward force of 11 N at a distance of 7.0 cm
from the axis of rotation.
• Find the net torque acting on
• center of mass.

47
Simple Machines
• A machine is any device that transmits or
modifies force, usually by changing the force
applied to an object.
• All machines are combinations or modifications of
six fundamental types of machines, called simple
machines.
• These six simple machines are the lever, pulley,
inclined plane, wheel and axle, wedge, and screw

48
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49
• Because the purpose of a simple machine is to
change the direction or magnitude of an input
force, a useful way of characterizing a simple
machine is to compare the output and input force.
• This ratio is called mechanical advantage.
• If friction is disregarded, mechanical advantage
can also be expressed in terms of input and
output distance.

50
In the first example, a force (F1) of 360 N moves
the trunk through a distance (d1) of 1.0 m. This
requires 360 Nm of work. In the second example,
a lesser force (F2) of only 120 N would be needed
(ignoring friction), but the trunk must be pushed
a greater distance (d2) of 3.0 m. This also
requires 360 Nm of work.
51
• The simple machines we have considered so far are
ideal, frictionless machines.
• Real machines, however, are not frictionless.
Some of the input energy is dissipated as sound
or heat.
• The efficiency of a machine is the ratio of
useful work output to work input.

The efficiency of an ideal (frictionless) machine
is 1, or 100 percent. The efficiency of real
machines is always less than 1.