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Gravity

Gravity is a natural phenomenon by which all things with mass are brought toward one another, including planets, stars and galaxies.

Direction of gravity on a planet

We say that gravity pulls things “down”.
If that were literally true, then gravity would pull objects down like this:

Yet would happen to people on the other side of the Earth?
They’d fall off!

falling-off-the-earth

Obviously this never happens. Why not?

Gravity only appears to pull down locally.

In the big picture, it pulls radially inward:

This is why we can walk around the world.

How does gravity affect the velocity of a falling object?

Gravity accelerates objects towards the center of the Earth
Thus, the velocity of a falling object picks up this much speed every second (*)

g =  9.81 m/s2   (more precise)

g =  10.0 m/s (good approximation)

(*) air resistance will change this. That’s a separate topic.

Here’s an example of how a falling body picks up speed, with the approximation.
Free Fall gravity 10 m s s

Here’s an example of how a falling body picks up speed, with a more precise number.

Free Fall gravity 9.8 m s s

This value is only true at the Earth’s surface – the value of g decreases as one gets further from the Earth’s surface.

The value of g is different on other planets and celestial bodies.

The Falling Apple

There is a well known story about how Newton discovered gravity – he was supposedly inspired by an apple falling on his head.  That’s balderdash. Everyone knew gravity existed, long before Newton. People also had a rough idea of the numerical value of the acceleration of gravity.

Newton didn’t discover gravity. Rather, he made the significant realization that gravity worked everywhere in the universe – not just on Earth.

He didn’t just hypotheisize this: He came up with an equation which could make predictions and be tested.

The Falling Moon

Newton hypothesized that the moon was a projectile, held in orbi tby Earth’s gravity.

This idea had never seriously been considered by anyone before then.

How could he show that this was reasonable?

Newton’s canon.

Dr. William Romanishin, emeritus professor of astronomy at the U. of Oklahoma, writes:

Imagine shooting a cannonball from a cannon parallel to the surface of the Earth from a high mountain. The faster the cannonball moves, the further it would travel before hitting the Earth. At a certain speed, the cannonball would fall around the curving Earth, and would come back and hit the cannon. Such a path is called an orbit.

If the cannonball were above the Earths atmosphere, so that there was no friction with the air, it would just go around and around forever. The cannonball is in “free fall”- it is falling around the Earth- but it never gets any closer to the surface! The cannonball (or Space Shuttle or whatever) does not need any further “push” once it gets into an orbit.

A simple way to think about what is going on is to think about tying a string around a rock and swinging it around in a circle. The string exerts a force on the rock which causes it to keep moving in the circle, rather than in a straight line. (If the string breaks, the rock flies off in a straight line.)

In an orbit, *gravity* provides the “string” that holds the orbiting body in place. So, a body in an orbit is constantly being pulled by gravity into a curved path. By Newton’s first law, a curved path requires a force- if there were no force then the object would move in a straight line.

http://hildaandtrojanasteroids.net/A1504-11feb11.html

Newton’s canon apps

http://waowen.screaming.net/revision/force&motion/ncananim.htm

http://interactives.ck12.org/simulations/physics/newtons-cannon/app/index.html

Newton's canon

Another way to show Newton’s canon

Newton’s apple-is-like-the-moon

Newton developed a formula for the gravitational attraction between two point sources.

How did Newton discover this law? How Newton discovered the law of gravity by Stephen Olson (PowerPoint) – Stephen L. Olsen, Professor of Physics, Department of Physics and Astronomy, University of Hawai’i at Manoa.

http://www.phys.hawaii.edu/~solsen/

Dr. David Stern has this presentation:  Newton’s theory of “Universal Gravitation”

On the same topic is this article: Newton’s Universal law of gravity: The Guardian article

Earth is falling around the Sun

The same is true for why planets revolve (orbit) around our Sun.

Who is orbiting whom?

People usually think of our Sun as standing perfectly still, and Earth (and other planets) revolve around it.

Yet both the Earth and Sun have mass. Both attract each other. For every force, there is an equal and opposite force (3rd law)

Thus the Earth (and other planets) pull on the Sun.

Wrong idea: Sun stands still, Earth revolves around it.

Better idea: Sun and Earth mutually rotate around the system’s center-of-gravity

Results?

  • Sun has a wobble

  • Since Sun is much more massive than Earth, center of mass is closer to the Sun

  • The center of mass is inside the Sun, so the resulting wobble caused by Earth is small.

What would this same system look like, as seen from the side?

What would this look like, if 2 objects of similar size orbited around each other?
The Pluto–Charon system

Pluto Charon barycenter

The Sun wobbles even more than this due to the pull from other worlds: Jupiter has a large mass, it causes a big wobble.

We can even use this wobble to detect the presence of planets around other stars. See One way to find a planet: Watch for a star’s wobble (NASA)

Law of Universal Gravitation

For any two objects, the force of attraction is proportional to the mass of each object.

The greater the masses, the greater the force of attraction.

Force decreases as the square of the distance

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

Hewitt, Conceptual Physics textbook

Hewitt, Conceptual Physics textbook

Let’s try this – plug in numbers

Hold 2 textbooks 1 meter apart from each other.

Average school textbook is about 1.5 kg  (3.3 pounds.)

How much gravitational force exists between them?

F = m1 x m2 / d2   = 1.5 x 1.5 / 1m2 = 2.25 Newtons (about half a pound)

But something doesn’t add up – half a pound of force?
We don’t see or feel that.

What about 2 students, standing one meter apart?

Students weigh 10 kilograms = 60 Newtons = about 120 pounds

F = m1 x m2 / d2   = 10 x 10 / 1 m2 = 100 Newtons (about 50 pounds pf attractive force)

Again something doesn’t add up – We don’t see or feel that.

Gravity isn’t equal to this relationship – it’s only proportional.

What is the proportionality constant?

Must be very small, since in daily life we never feel the gravitational attraction of people, cars, or even of large buildings.

That is what “big G” is for

G = Universal Gravitational Constant

G describes the strength of gravity in our universe.

Originally measured by Henry Cavendish in 1798.

How did he do this?  Cavendish Experiment to Measure Gravitational Constant .

Cavendish showed that the strength of gravity is very small:

Universal Gravitational constant

Whoah… times ten raised to the negative eleventh power.

Decimal moves eleven places to the left:   0.0000000000667 N•m2/ kg2

This matches our everyday experience:
When you stand a few inches away from a 3,000 pound car, do you feel the car’s gravity pulling you?
Nope. Gravity between a person and car is very small.

When you go into Boston, and walk by a tall building, do you feel the building pulling on you?
Nope. Gravitational attraction between the building and you is real,
but so small that you wouldn’t notice.

Here’s the equation:

Notice that it is an inverse-square rule

G = Universal gravitational constant

m1 = mass of object 1

m2 = mass of object 2

r = distance between the objects

The Inverse Square Law

The strength of gravity is reduced with distance. How?
Consider cooking spray.  Oil is sprayed through an opening.
In this image, the cooking spray hits a piece of toast and deposits an even layer of butter, 1 mm thick.

Hewitt textbook

Hewitt textbook

Now put the toast twice as far from the butter gun.
Now the butter covers 2X as much toast vertically and horizontally.
That’s 4X the area.

Since the oil has been diluted, to cover 4X as much area,
its thickness will be 1/4 as much, or 0.25 mm.

When the butter gets twice as far, it becomes only 1/4 as this.

If it travels 3 times as far, it will spread out to cover 3 x 3,
or 9, pieces of toast.

So now the butter will only be 1/9th as thick.

(1/9 is the inverse square of 3)

This pattern is called an inverse-square law.

The same is true for a can of spray paint: as the paint travels further, it covers a wider area, so the paint per area is inversely less thick.

Hewitt Conceptual Physics worksheets

Hewitt Conceptual Physics worksheets

The same is true for anything that spreads out – including gravity.

Don’t confuse g and G

G = Universal gravitational constant
= the strength of the gravitational force in our universe.
This is the same everywhere in the universe.

g = local acceleration of gravity

     Acceleration on the surface of a particular planet or moon.

Is there gravity in space?

As we get further from the Earth’s surface, the gravity at that altitude is weaker.

Notice that the apple is far out in space – yet it still feels some gravity from Earth! Even a thousand miles from Earth, there is some gravity there.

So why are astronauts weightless when they orbit the Earth in the international space station?

It can’t be because they are “in space” … as the image shows, there is plenty of gravity, even thousands of miles out in space.

They must be weightless for a different reason.

Gravitational Field

We’ll be mentioning a gravitational fieldWhat are fields?

A planet is surrounded by a gravitational force field.
It interacts with objects, and causes them to experience gravitational forces.

When a comet travels through the solar system, it feels a pull from the Earth, even if it never touches the Earth.
How can the comet, millions of miles away, feel Earth’s pull without  contact? It is in our gravitational field.

You already know about the magnetic field of a magnet: Iron filings sprinkled over a sheet of paper

Or place a bunch of small compasses around a bar magnet – this shows us the existence of a field.

https://kaiserscience.wordpress.com/physics/electromagnetism/magnetism/

The Earth is a giant magnet, with a magnetic field reaching out into space.

Earth North Geographic Pole South Magnetic Pole

Extend this idea to gravity: Earth has a gravitational field:

Gravity inside a planet?

Imagine a hole drilled through Earth, from the North Pole to the South Pole. If you fell in at the North Pole, then you’d fall and gain speed all the way down to the center. Then you would continue moving, and lose speed all the way to the South Pole.

You gain speed moving toward the center, and lose speed moving away.

Without air drag, the trip would take nearly 45 minutes. If you failed to grab the edge, then you’d fall back toward the center, overshoot, and return to the North Pole in the same amount of time.

At the beginning of the fall, your acceleration would be g =  10 m/s2

Yet this acceleration gets less as we move towards the center

Why? Because as you move toward Earth’s center, you are now also pulled “upward” by the part of Earth that is “above” you.

When you get to the center of Earth, the pull “down” is balanced by the pull “up.”

You are pulled in every direction equally, so the net force on you is zero.

The gravitational field of Earth at its center is zero.

Gravity inside a planet

_________________________

Force gravity = (G•M•m•r) / (REarth)3       For r < R Earth

Force gravity = (G•M•m) / r2                     For r >= REarth

Gravity Inside a Sphere

Ocean tides

How does gravity create ocean tides?

LAB: Why is there a tidal bulge opposite the moon?

 

Newton’s 3rd law of motion: Forces come in pairs

Orbits

In order for an object to remain in a stable circular orbit, about a larger body, the gravitational force must balance the centripetal force.

Newton’s Second Law: F=ma

Insert gravitational force for F, and centripetal acceleration for a

Derive equation for the tangential velocity vt (orbital speed) of the orbiting body, m2.

Note: gravitational force on m2 is directed inward toward the center.

Centripetal acceleration of m2 is also always directed inward

Thus gravitational force and centripetal acceleration of m2 are always same sign.

https://www.wyzant.com/resources/lessons/science/physics/gravitation

Galileo

PBS NOVA Galileo’s Battle for the Heavens
Although Newton discovered the law of gravity nearly 300 years ago, until Einstein came along, scientists had no idea how gravity actually worked. View the “Newton’s Embarrassing Secret” segment. (6 minutes)

Einstein’s success in explaining gravity as warps and curves in the fabric of space and time set him on a quest to unify gravity with electricity and magnetism. View the “A New Picture of Gravity” segment (7 minutes) from “The Elegant Universe” (NOVA)  Gravity, from PBS The Elegant Universe

Other articles here on KaiserScience

Gravitational repulsion and the Dipole Repeller

Apps

Gizmos Gravity physics apps HTML5

https://phet.colorado.edu/sims/my-solar-system/my-solar-system_en.html

http://lasp.colorado.edu/education/outerplanets/orbit_simulator/

https://academo.org/demos/orbit-simulator/

http://astro.unl.edu/classaction/animations/renaissance/kepler.html

http://www.windows2universe.org/physical_science/physics/mechanics/orbit/orbit_shape_interactive.html

https://phet.colorado.edu/en/simulation/gravity-and-orbits

Newton’s canon

http://waowen.screaming.net/revision/force&motion/ncananim.htm

http://interactives.ck12.org/simulations/physics/newtons-cannon/app/index.html

PowerPoint presentations

AP Physics: Gravity and Circular Motion Chap. 5 Giancoli

Conceptual Physics: Universal Gravitation Chap 13 Hewitt

Chap 13 Hewitt Chapter on gravity complete text

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

8.MS-ESS1-2. Explain the role of gravity in ocean tides, the orbital motions of planets, their moons, and asteroids in the solar system.

HS-PS2-4. Use mathematical representations of Newton’s law of gravitation and Coulomb’s law to both qualitatively and quantitatively describe and predict the effects of gravitational and electrostatic forces between objects.

Next Generation Science Standards

HS-PS2.B.1 ( High School Physical Sciences ): Newton’s law of universal gravitation and Coulomb’s law provide the mathematical models to describe and predict the effects of gravitational and electrostatic forces between distant objects.

A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)

PS2.B: TYPES OF INTERACTIONS
Gravitational, electric, and magnetic forces between a pair of objects do not require that they be in contact. These forces are explained by force fields that contain energy and can transfer energy through space. These fields can be mapped by their effect on a test object (mass, charge, or magnet, respectively). Objects with mass are sources of gravitational fields and are affected by the gravitational fields of all other objects with mass. Gravitational forces are always attractive. For two human-scale objects, these forces are too small to observe without sensitive instrumentation. Gravitational interactions are non-negligible, however, when very massive objects are involved. Thus the gravitational force due to Earth, acting on an object near Earth’s surface, pulls that object toward the planet’s center. Newton’s law of universal gravitation provides the mathematical model to describe and predict the effects of gravitational forces between distant objects.

Learning Standards: Common Core Math
All the math standards necessary for success in this chapter come from Common Core 7th grade math content.
CCSS.MATH.CONTENT.7.EE.B.4  Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
CCSS.MATH.CONTENT.8.EE.C.7  Solve linear equations in one variable
CCSS.MATH.CONTENT.HSA.SSE.B.3  Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (including isolating a variable)
CCSS.MATH.CONTENT.HSA.CED.A.4  Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Standards related to understanding the inverse-square law
CCSS.Math.Content.7.RP.A.2a ( Grade 7 ): Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

CCSS.Math.Content.7.RP.A.2c ( Grade 7 ): Represent proportional relationships by equations.
CCSS.Math.Content.7.RP.A.3 ( Grade 7 ): Use proportional relationships to solve multistep ratio and percent problems.
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