Universal Gravitation

Chapter 12

The Falling Apple

• The idea that gravity extends through the universe is attributed to Sir Issac Newton

• He knew that if an object undergoes a change in direction or speed, a force is responsible

• Newton had the insight to figure out that the moon orbits Earth because it is falling towards Earth, avoiding its straight-line path

The Falling Apple

The Falling Moon

• The moon falls in the sense that it falls beneath the straight line it would follow if no force acted on it

• Newton imagined firing a cannonball from a mountain top. If he fired the cannonball with a small horizontal speed, it would follow a parabolic path and hit Earth soon. If he fired the same cannonball with greater horizontal speed, it would go farther and have a less curved path. If he fired the cannonball with great enough horizontal speed, he reasoned that the cannonball’s path would become a circle and it would circle indefinitely (orbit)

The Orbital Cannon

Launch Speed less than 8000 m/s Projectile falls to Earth

                                             

Launch Speed less than 8000 m/s Projectile falls to Earth

                                             

Launch Speed equal to 8000 m/s Projectile orbits Earth - Circular Path

                                             

Launch Speed greater than 8000 m/s Projectile orbits Earth - Elliptical Path

                                           

Cannonball in Orbit

Newton’s Law of Universal Gravitation

• Law of Universal Gravitation – every object attracts every other object with a force that for any two objects is directly proportional to the mass of each object, the force also decreases as the square of the distance between the centers of mass of the objects increases

• The farther away the objects are from each other, the less the force of attraction between them

The Universal Gravitational Constant, G

• When the proportionality constant, G, is introduced, we can write the Law of Universal Gravitation in the form of an equation:

F = G(m1)(m2)/d2

• G = 6.67 x 10-11 N∙m²/kg²• We only sense gravitation for bodies the

size of a planet, smaller objects we can’t sense, but the force is there!

Law of Universal Gravitation

The Inverse-Square Law

• When a quantity varies as the inverse square of its distance from its source, it follows an inverse-square law

• The greater the distance from Earth’s center, the less an object will weigh

• You may weigh 300N at sea level, but only 299N at the top of Mount Everest

Inverse-Square Law

Universal Gravitation

• Earth is round due to gravitation, it has “attracted” itself together to create a spherical shape

• When planets deviate from their normal orbits, they have been pulled due to the attraction to other planets, this deviation is called perturbation

• This is how we found Neptune, Uranus was acting strangely, in a way that could only be accounted for by another planet perturbing its orbit

Perturbation

Gravitational Interactions

Chapter 13

Gravitational Fields• A force field exerts a force on any objects in its

vicinity• A magnetic field shows the strength of magnetism

around an object, an electric field show the strength of electricity, etc.

• Gravitational Field – a force field that exists in the space around every mass or group of masses

• We represent fields with imaginary field lines; the closer the lines are together, the stronger the field

• The strength of Earth’s gravitational field is the force per unit mass exerted by Earth on any object

g = F/m = 9.8 N/kg = 9.8 m/s²• We can verify this number by using the law of

universal gravitation and known values for Earth (it is solved out in your textbook!)

Gravitational Fields

Gravitational Field Inside a Planet

• The gravitational field of Earth exists inside, as well as outside

• If you dug a tunnel from the North Pole to the South Pole, you would gain speed all the way to the center, and then continue going to the South Pole while losing speed

• At the beginning of your trip, your acceleration would be g; but by the middle of your trip, you would have no acceleration (it has decreased going towards the center to 0)

• Why?• As you are being pulled down by the center of

Earth, you are also being pulled UP by the surface (in the center, you are being pulled in all directions equally!)

Gravitational Field Inside a Planet

Weight and Weightlessness• The force of gravity, like any force, causes an

acceleration• The gravitational force between you and Earth pulls

you against the floor; by Newton’s 3rd Law, the floor pushes up on you

• When you stand on a scale, it is reading these two combined forces

• If I repeated this in a moving elevator, my weight would change during any accelerated motion (upwards, weight goes up!)

• Now, take away the support force of the floor pushing up, and I would constantly be in free fall, weightless!

• Weightlessness – the condition of free fall toward or around Earth, in which an object experiences no support force (and exerts no force on a scale)

Weight and Weightlessness

Ocean Tides and the Moon• Ocean tides are caused by differences in the

gravitational pull of the moon on opposite sides of Earth

• The moon’s attraction is greater on Earth’s oceans closer to the moon

• Because Earth and the moon orbit each other around their shared center of mass (a point inside Earth about ¾ from the center), we get two sets of tides a day (high and low)

• Because of the mutual attraction between Earth and the moon, both have a pair of “tidal bulges” on them

Ocean Tides and the Sun• The sun also contributes to ocean tides, but since it is further

away, it influences tides to a lesser extent• Spring Tides – A high or low tide that occurs when the sun,

Earth and the moon are all lined up so that the tides due to the sun and the moon coincide, making the high tides higher than average and the low tides lower than average

• Lunar Eclipse – The cutoff of light from the full moon when Earth is directly between the sun and the moon, so that Earth’s shadow is cast on the moon

• Solar Eclipse – The cutoff of light from the sun to an observer on Earth when the moon is directly between the sun and Earth

• Neap Tides – A tide that occurs when the moon is halfway between a full moon and a new moon, in either direction. The tides due to the sun and the moon partly cancel, so that the high tides are lower than average and the low tides are not as low as average

Ocean Tides

Black Holes• A star is so large that it must constantly produce an outward

force to stop gravity from collapsing the star, it gets this energy from nuclear fusion (the fusing together of hydrogen atoms to create helium)

• When a star runs out of hydrogen, gravitation will take over and the star will begin to collapse in on itself

• For very massive stars (much larger than our sun) this process may result in the star crushing itself out of existence and forming a black hole

• Black Hole – a mass that has collapsed to so great a density that its enormous local gravitational field prevents light from escaping

• Although black holes cannot be seen, we can see their effects; mass emitting x-rays as it accelerates to nothingness

• Most astrophysicists believe that near the centers of most galaxies is a black hole, causing the entire galaxy to rotate

Black Holes

Satellite Motion

Chapter 14

Earth Satellites• An Earth satellite is a projectile that falls around Earth

rather than into it• The surface of Earth drops a vertical distance of ~5

meters for every 8000 meters tangent to its surface; meaning, if you were in the ocean you would only see the top of a 5-m-tall mast on a ship that is 8 kilometers away

• If you threw a ball fast enough to go a horizontal distance of 8 kilometers during the time (1 second) it takes to fall 5 meters, it would follow the curvature of Earth!

• This would be about 29,000 km/hr (or 18,000 mi/hr)• At this speed atmospheric friction would burn any

object to a crisp, that’s why satellites must be at 150 kilometers or more above Earth’s surface

Earth Satellites

Circular Orbits• In a circular orbit, the speed of a circling

satellite is not changed by gravity• A satellite in a circular orbit around Earth is

always moving perpendicular to gravity and parallel to Earth’s surface at a constant speed

• For a satellite close to Earth, the time for a complete orbit around Earth, its period, is about 90 minutes

• The higher the altitude, the lower the orbital speed, and the longer the period

• Communication satellites orbit at about 6.5 Earth radii from Earth’s center, so that their period is 24 hours (they will always stay above the same location)

Circular Orbits

Elliptical Orbits• A projectile above the atmosphere moving at a

horizontal speed greater than 8km/s will overshoot a circular path and trace an oval-shaped path – an ellipse

• Ellipse – an oval-shaped curve that is the path of a point that moves such that the sum of its distances from two fixed points (foci) is constant

• Satellite speed, which is a constant for a circular orbit, will vary in an elliptical orbit

• When the satellite moves away from Earth, it loses speed (it’s working against gravity)

• As it comes back towards Earth, it gains speed (it’s working with gravity)

• The satellite then rejoins its path with the same speed it started with!

Elliptical Orbits

Energy Conservation & Satellite Motion

• Remember, moving objects have kinetic energy (KE) and an object above Earth’s surface has potential energy (PE)

• A satellite has both KE and PE, such that the sum of the KE and PE everywhere is constant

• In a circular orbit, the distance between a planet’s center and the satellite never changes (PE), therefore, by conservation of energy, its kinetic energy is also constant

• In an elliptical orbit, both speed and distance change• Apogee – the point in a satellite’s elliptical orbit

farthest from the center of Earth (PE is the greatest)• Perigee – The point in a satellite’s elliptical orbit

where it is nearest the center of Earth (KE is the greatest)

Energy Conservation & Satellite Motion

Escape Speed• In order to put an object into orbit around Earth speed

and direction are important• Remember, “what goes up must come down”; if I shot

a rocket straight up at 8 km/s, it would back down at 8 km/s – instead I must shoot it horizontally

• Neglecting air resistance, fire anything at any speed greater than 11.2 km/s, and it will leave Earth, going more and more slowly, but never stopping (this is the value of the escape speed)

• Escape Speed – the minimum speed necessary for an object to escape permanently from a gravitational field that holds it

• The larger the object, the greater the escape speed (the Sun’s is 42.2 km/s)

• In order for the probe Pioneer 10 to escape the Sun’s gravitational field NASA scientists used Jupiter’s gravitational field to help it gain enough speed

Escape Speed

Assignment

• Read Chapters 12-14 (pg. 168-211)

• Ch. 12: Do #21-35 (pg. 181), App. F #1-6 (pg. 676)

• Ch. 13: Do #25-38 (pg. 197-198), App. F #1-2 (pg. 677)

• Ch. 14: Do #15-28 (pg. 211), App. F #1-2 (pg. 677)