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Rotational Motion and The Law of Gravity

- Ch 7

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.

- The Ferris wheel turns about an axis.
- The Ferris wheel rotates, while the riders

revolve about its axis. - 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.

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

Centripetal Acceleration

- Tangential speed (vt) depends on distance
- When tangential speed is constant, motion is

described as uniform circular motion

- 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

Centripetal Acceleration

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

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

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

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.

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

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?

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.

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.

- This original drawing by Isaac Newton shows how a

projectile fired fast enough would fall around

Earth and become an Earth satellite.

- 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.

The gravitational force attracts Earth and the

moon to each other. According to Newtons 3rd

Law.

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

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

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

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.

- 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.

- When G was first measured in the 1700s,

newspapers everywhere announced the discovery as

one that measured the mass of Earth

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

- 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

- Field lines represent the gravitational field

about Earth

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

mass and radius - Your weight is less at the top of a mountain

because you are farther from the center of Earth.

- 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.

- The two tidal bulges remain relatively fixed with

respect to the moon while Earth spins daily

beneath them.

- Earths tilt causes the two daily high tides to

be unequal.

Keplers Laws of Planetary Motion

- Newtons law of gravitation was preceded by

Keplers laws of planetary motion. - Keplers laws of planetary motion are three

important discoveries about planetary motion made

by the German astronomer Johannes Kepler.

- 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.

- 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.

- 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.

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.

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

- 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.

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?

- 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.

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.

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

- 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.

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.

- 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

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
- the ball about its
- center of mass.

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

(No Transcript)

- 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.

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.

- 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.