Chapter 8:Universal

Gravitation

Chapter 8 Objectives

Relate - Kepler’s laws of planetary motion to Newton’s Laws of universal gravitation.

Calculate - Periods and speeds of orbiting objects.

Describe - Cavendish’s method for measuring the universal gravitation constant (G).

Concept Development Map

Gravity

Weight, heavy(gravis)

Grave (serious; sober)

Importance, significance

DefinitionsExamples/Applications

Falling Objects

Satellite Orbit

Weightlessness

Flight, Ballooning

Amusement Rides

What is it?

Sports

Bouncing; Bungie

Attraction, Pull, Fall, Drop, Move, towards center of earth

Slowing Trucks; objects

8.1 Motions in the

Heavens and on Earth

Johannes Kepler (1571-1630)

Johannes Kepler became an assistant to Tycho Brahe (1546-1601) in Prague. Brahe had very precise data on planetary motion but believed that the earth was the center of the universe. Kepler wanted to use a sun-centered model of planetary motion along with Brahe’s data. This led to Kepler’s three laws of planetary motion.

Kepler’s Three Laws of Planetary Motion

Law 1: The Paths of the Planets are elliptical, with the sun at one focus.

Discussion: Why don’t we feel really cold as an entire planet when we are far from the sun?

Kepler’s Three Laws of Planetary Motion

Law 2: Imaginary line from sun to planet sweeps out equal areas in equal time, so planets move faster when close to sun and slower when farther away.

eliptical motion animation

Discussion: Why not feel accelerations when we speed up and slow down in the seasons?

Kepler’s Three Laws of Planetary Motion

Law 3: Square of the ratio of the periods of any two planets about the sun is equal to the cube of the ratio of their average distance from the sun.

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⎝ ⎜

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Kepler’s Laws of Planetary Motion

Newton’s Law of Universal Gravitation

In 1666, Isaac Newton knew that a force kept the planets in orbit. To follow Kepler’s laws the magnitude of the force must vary inversely with the square of the distance between their centers and must be proportional to the masses of the two planets. This combined into:

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F =GmAmBd2

The constant of proportionality turned out to be a universal number (capital G) that holds anywhere in the universe.

Little g versus Big G

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F =GmAmBd2

= mAgmearth = 5.974×1024 kg

dearth = rearth = 6,378 km

G = 6.673 × 10-11 m3 kg-1 s-2

5.974(10)24 kg 6.673(10)-11 m3 (1 km)2

(6.378(10)3 km)2 kg s2 (103 m)2 = 9.8 m/s2 !!!!!

Newton’s Law of Universal Gravitation

The Inverse Square Law is Everywhere

Cavendish’s Measurement of “G”

In 1798, Englishman Henry Cavendish (1731-1810) used an apparatus similar to the one below to measure the universal gravitation constant G.It had taken over a hundred years before someone could measure it accurately. A metal rod was attached to a ceiling support and the twisting of the metal caused a measurable force.

Von Jolly: Measuring “G” More Accurately

Discovering Neptune

8.1 Vocabulary

Gravitational Force: The attractive force that exist between all objects.

Law of Universal Gravitation: Gravitational force between any two objects is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centers.

8.2 Using the Law of

Universal Gravitation

Satellite Velocity

CA Standard 1.f. and 1.g.

g. Students know applying a force to an object perpendicular to the direction of its motion causes the object to change direction but not speed (e.g., Earth's gravitational force causes a satellite in a circular orbit to change direction but not speed).

f. Students know circular motion requires the application of a constant force directed toward the center of the circle.

Satellite Velocity

Centripetal Force and acceleration.

ac = v2/r

Centripetal acceleration always points to the center of the circle or ellipse. Its magnitude is equal to the square of the speed, divided by the radius of motion.

Centripetal force is mass times the centripetal acceleration.

Fc = mv2/r

Satellite Velocity

Centripetal Acceleration:

F = ma ac = v2/r

Newton’s Second Law:

Force Centripetal:

Fc = mv2/r

Satellite in Free Fall

Figure 8.6: Satellites are in a state of constant free fall.

Satellite Velocity

Newton’s Inverse Square Law:Force Centripetal:

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F =GmAmBd2

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Fc =mv 2

r

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mv 2

r=G

mAmBd2

Combining:

Satellite Velocity

Solving for velocity:

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mSv2

r=G

mEmSr2

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v 2 =GmEr

Or

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v = GmEr

Satellite Velocity

Solving for velocity:

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v = GmEr

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v =Δd

Δt=

2πr

Τ= G

mEr

But:

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Τ =2πr3

GmE

Leads to:

Weight and Weightlessness

Weightlessness

On that day, the physicist's daydream ended with what he later called his "happiest moment." He surmised that the unlucky painter would feel weightless when accelerating toward the ground. This clue led Einstein to reason that gravity and acceleration must be equivalent. Called the "equivalence principle," this idea was the seed that - over the next nine years - bloomed into Einstein's masterpiece, the "General Theory of Relativity." This new theory laid the foundation for relativistic astrophysics and modern cosmology.

“If a person falls freely, he won’t feel his own weight. This simple thought made a deep impression on me.” - Albert Einstein.

Apparent Weightlessness

Two Kinds of Mass

Gravitational Mass:Inertial Mass:

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minertial =Fneta

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mgravitational =Fgravg

=r2FgravGmE

Intertial Mass: Truck accelerates on level ground. Block of ice slides to back of the truck bed.

Gravitational Mass : Truck goes with constant velocity up a mountainous slope. Again the block of ice slides to back of the truck bed. It is pulled to the earth’s center.

Einstein’s Theory of Gravity

Einstein’s used his General Theory of Relativity to explain the effect of gravity. The concept of gravitational field allows us to picture the way gravity acts on bodies far away. However, neither theory explains the origin of gravity. According to Einstein, gravity is an effect on space itself. One way to picture it is to place round masses on a sheet of rubber or elastic fabric. The greater the mass, the more it makes a depression (indentation) in the fabric. If an small object gets close to the large depression, then its motion will be deflected.

But don’t forget that this “Plane” can be rotated in any direction!!

8.2 Vocabulary

Gravitational Mass: Ratio of gravitational force exerted on an object to its acceleration due to gravity.

Inertial Mass: Ratio of net force exerted on an object to its acceleration. The mass of an object as measured by its resistance to acceleration.

7.3 Circular Motion

Circular Motion

Centripetal Force: A force that acts towards the center of the circle of motion. From the Latin centri (center) + petere (to move toward).

Centrifugal Force: A force that acts away from the center of the circle of motion. From the Latin centri (center) + fugere (to flee).

Circular Motion

Centripetal Acceleration:

a = v2/r

F = ma = m (v2/r)

Centripetal Force

Rotational Speed

Circular Motion

Torque

Torque (Latin: torquere, to twist): A force that is applied perpendicular to the lever arm that results in rotational motion around an axle. A tangential force multiplied by the radius of the part it rotates.

= (Fcosl

What angle gives maximum torque?What angle gives minimum torque?

Torque in Balance: Seesaw

The torque is balanced on both sides of the seesaw. They are equal and opposite (Counter-clockwise vs. Clockwise ).

=

m1g l1= m2g l2

F1 l1= F2 l2