Circular Motion and Gravitation

Holt

Chapter 7

Honors Physics

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Circular Motion and GravitationChapter 7

Table of Contents

Section 1 Circular Motion

Section 2 Newton’s Law of Universal Gravitation

Section 3 Motion in Space

7.1 Circular Motion

Any object that revolves about a single axis undergoes circular motion.

7.1 Circular Motion

Tangential speed (vt):

speed of an object along an imaginary line drawn tangent to the object’s circular path

depends on an object’s distance from the center of the circular path

is constant in uniform circular motion

7.1 Circular Motion Centripetal Acceleration (ac):

Tangential acceleration is due to a change in speed.

• due to a change in direction

• is directed toward the center of the circle

ac =vt

r

2

Chapter 7Centripetal Acceleration

Acceleration is a change in velocity.

(a) As the particle moves from A to B, the direction of the particle’s velocity vector changes.

(b) For short time intervals, ∆v is directed toward the center of the circle.

Centripetal acceleration is always directed toward the center of a circle.

Section 1 Circular Motion

7.1 Circular Motion

Chapter 7Centripetal Acceleration REPEAT

Centripetal acceleration results from a change in direction.

In circular motion, an acceleration due to a change in speed is called tangential acceleration.

A car traveling in a circular track can have both centripetal and tangential acceleration. Because the car is moving in a circle, the car has a

centripetal component of acceleration. If the car’s speed changes, the car also has a

tangential component of acceleration.

Section 1 Circular Motion

7.1 Circular Motion Centripetal Force (Fc): the net force

directed toward the center of an object’s path Centripetal means center seeking.

Fc =mvt

r

2

Fc = mac

Centripetal force overcomes the path of inertia. Inertia is not a force.

Fc and ac are in the same direction. The centripetal force is in the plane of the object and perpendicular to the tangential speed of the object.

Chapter 7Centripetal Force Consider mass m that is being whirled in a horizontal

circular path of radius r with constant speed. The force exerted by the string has horizontal and vertical

components. The vertical component is equal and opposite to the gravitational force. Thus, the horizontal component is the net force.

This net force, which is directed toward the center of the circle, is a centripetal force.

Chapter 7Centripetal Force

• If the centripetal force vanishes, the object stops moving in a circular path.

• A ball that is on the end of a string is whirled in a vertical circular path.– If the string breaks at the position shown

in (a), the ball will move vertically upward in free fall.

– If the string breaks at the top of the ball’s path, as in (b), the ball will move along a parabolic path.

7.2 Newton’s Law of Universal Gravitation

Gravitational Force Orbiting objects are in freefall.

Chapter 7Gravitational Force The centripetal force that holds the planets in

orbit is the same force that pulls an apple toward the ground. It is the gravitational force.

Gravitational force is the mutual force of attraction between particles of matter.

The amount of gravitational force depends on the masses of the objects and on the distance between them.

Section 2 Newton’s Law of Universal Gravitation

7.2 Newton’s Law of Universal Gravitation

Gravitational Force

Fg = Gm1m2

r 2

G = 6.673 x 10-11 N.m2/kg2 G is the constant of universal gravitation.r = the distance between the centers of the two masses, m1 and m2.

Fg ~ m1m2

r 2

rm1

m2

Chapter 7Newton’s Law of Universal Gravitation

• The gravitational forces that two masses exert on each other are always equal in magnitude and opposite in direction.

• This is an example of Newton’s third law of motion.• One example is the Earth-moon system. • As a result of these forces, the moon and Earth each orbit the center of

mass of the Earth-moon system. Because Earth has a much greater mass than the moon, this center of mass lies within Earth.

7.2 Newton’s Law of Universal Gravitation Gravitational Force

The tides result from the difference between the gravitational force at Earth’s surface and at Earth’s center.

Spring tides are higher high and lower low tides than normal.

Neap tides are lower high and higher low tides than normal.

NOAA's National Ocean Service: Animation of spring and neap tides

7.2 Newton’s Law of Universal Gravitation Gravitational Force

Henry Cavendish, 1798, determined the value of G,G = 6.673 x 10-11 Nm2/kg2

and then he determined ME.

Chapter 7

Applying the Law of Gravitation, continued

weight = mass gravitational field strength Because weight depends on gravitational field

strength, weight changes with location:

Section 2 Newton’s Law of Universal Gravitation

weight = mg

g Fg

m

GmmE

mr2

GmE

r2

• On the surface of any planet, the value of g, as well as your weight, will depend on the planet’s mass and its radius.

7.2 Newton’s Law of Universal Gravitation

Weight changes with location.

Fg = Gm1m2

r 2

Fg = W = m1g

m1g = Gm1ME

r 2

g = GME

r 2

Gravitational Field Strength

7.2 Newton’s Law of Universal Gravitation Gravitational Force is a field force.

A gravitational force is an interaction between a mass and the gravitational field created by other masses.

Gravitational potential energy is stored in the gravitational field. Gravitational field strength is g = Fg/m and equals free-fall acceleration. Gravitational field strength rapidly decreases as the distance from Earth

increases.

7.2 Newton’s Law of Universal Gravitation

Gravitational mass and Inertial mass are the same.

Newton’s second law of motion gives inertial mass (amount of matter in an object).

Newton’s law of universal gravitation gives gravitational mass (amount of attraction objects have for each other).

Fg = Gm1m2

r 2

F = ma

7.3 Motion in Space

Chapter 7Kepler’s Laws (1609, 1619)

Kepler’s laws describe the motion of the planets.

First Law (The Law of Ellipses): Each planet travels in an elliptical orbit around the sun, and the sun is at one of the focal points.

Section 3 Motion in Space

Chapter 7Kepler’s Laws

Thus, the planet

travels faster when it

is closer to the sun

and slower when it is

farther away.

Section 3 Motion in Space

• Second Law (The Law of Equal Areas): An imaginary line drawn from the sun to any planet sweeps out equal areas in equal time intervals. 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. Planets move faster closer to the sun.

Chapter 7Kepler’s Laws (1609, 1619)

Third Law: Kepler's third law - sometimes referred to as the law of harmonies - compares the orbital period and radius of orbit of a planet to those of other planets. The square of a planet’s orbital period (T2) is proportional to the cube of the average distance (r3) between the planet and the sun.

Section 3 Motion in Space

PlanetPeriod

(s)AverageDist. (m)

T2/R3

(s2/m3)

Earth 3.156 x 107 s 1.4957 x 1011 2.977 x 10-19

Mars 5.93 x 107 s 2.278 x 1011 2.975 x 10-19

The Law of Harmonies

PlanetPeriod

(yr)Ave.

Dist. (au)T2/R3

(yr2/au3)

Mercury 0.241 0.39 0.98

Venus .615 0.72 1.01

Earth 1.00 1.00 1.00

Mars 1.88 1.52 1.01

Jupiter 11.8 5.20 0.99

Saturn 29.5 9.54 1.00

Uranus 84.0 19.18 1.00

Neptune 165 30.06 1.00

Pluto 248 39.44 1.00

Chapter 7

Kepler’s Laws Kepler’s laws were developed a

generation before Newton’s law of universal gravitation (1687).

Newton demonstrated that Kepler’s laws are consistent with the law of universal gravitation.

The fact that Kepler’s laws closely matched observations gave additional support for Newton’s theory of gravitation.

Chapter 7

Kepler’s Laws, continued Kepler’s third law states that T2 r3.

The constant of proportionality is 42/Gm, where m is the mass of the object being orbited.

So, Kepler’s third law can also be stated as follows:

22 34

T rGm

Section 3 Motion in Space

Chapter 7Kepler’s Laws, continued

Kepler’s third law leads to an equation for the period of an object in a circular orbit. The speed of an object in a circular orbit depends on the same factors:

T 2r3

Gm vt G

m

r

• Note that m is the mass of the central object that is being orbited. The mass of the planet or satellite that is in orbit does not affect its speed or period.

• The mean radius (r) is the distance between the centers of the two bodies.

Section 3 Motion in Space

Chapter 7

Planetary Data

Section 3 Motion in Space

Chapter 7Weight and Weightlessness

To learn about apparent weightlessness, imagine that you are in an elevator: When the elevator is at rest, the magnitude of the

normal force acting on you equals your weight. If the elevator were to accelerate downward at 9.81

m/s2, you and the elevator would both be in free fall. You have the same weight, but there is no normal force acting on you.

This situation of no normal force is called apparent weightlessness.

Astronauts in orbit experience apparent weightlessness.

Section 3 Motion in Space

Chapter 7Weight and Weightlessness

Section 3 Motion in Space

The gravitational fields of planets are used to direct the travel (paths) of space probes.