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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 often say that gravity pulls things “down”. Yet if that were true then gravity would pull objects down like this:

If that were true then what 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? Because gravity only appears to pull down locally. It actually pulls radially inward:

This is why we can walk around the world.

Earth has an inward point gravitational field. What is a gravitational field?

How does gravity affect the velocity of a falling object?

This section is about what happens here on Earth. These numbers will be different on other planets, because other worlds have different levels of gravity.

Here on Earth, by experiment we observe that falling objects pick up (about) 10 meters/second of speed, every second.

This can be written as 10 m/s/s  or  10.0 m/s2

We say that Earth’s acceleration of gravity = g =  10.0 m/s2

“10” is a nice, easy to use number. We can use this as a decent estimate.

More precisely the acceleration is g =  9.81 m/s2

Even that number is an approximation. Earth isn’t exactly spherical, and the distribution of mass inside the Earth isn’t the same everywhere. Therefore the acceleration of gravity is slightly higher or lower over some places on Earth. And yes, that difference really matters, especially for GPS satellites.

Examples

Here’s an example of how a falling body picks up speed, approximating g as 10.0 m/s2 

Here’s another example of how a freely falling body picks up speed.

Here g is more precisely approximated as 9.81 m/s2

Now we make things slightly more complicated – yet slightly more accurate.

g is only 9.81 m/s2 near the Earth’s surface.

This number decreases as one gets further from the Earth’s surface. (Gravity gets weaker when we get away from a planet.)

And of course the value of g is different on other planets and celestial bodies.

The story of Newton and 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 total balderdash. Everyone knew that gravity existed, long before Newton.

So what about gravity did Newton really discover? He made the significant realization that gravity worked everywhere in the universe – not just on Earth. And he found a simple algebraic law to express this. And this one, simple law, that can be taught even in 9th grade, can explain the motion of objects across the world, across the solar system, and across our galaxy!

How did he get this idea? He spent years observing and doing math about the motion of all sorts of objects. Objects on the earth, and the motions of objects he observed in the heavens. And one day he asked a question – is the moon a projectile, held in orbit by Earth’s gravity? If so, what would that mean?

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

How could he show that this was reasonable?  He developed the argument we today call 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. A

t 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

For a more in-depth look at this see How Newton discovered the law of gravity by Stephen Olson (PowerPoint) By Stephen L. Olsen, Professor of Physics, Department of Physics and Astronomy, University of Hawai’i at Manoa.

Or see Newton’s theory of “Universal Gravitation”

And 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.

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.

What is weight?

Weight is a force created by gravity pulling an object downwards.

This Canadian hipster is holding a KaiserScience laptop.

Gravity tries to accelerate the laptop downwards; that creates a downward force, i.e. weight.

Here, that force is applied to his hand.

Can we convert between weight and mass?

In general, no. Mass is always the same, while weight depends on how much gravity the object experiences. My mass is the same everywhere, but my weight would be less on the moon and more on Mars.

But as long as we confine our example to comparisons made on the same planet, then in this limited case, yes.

The gravity of Earth imparts a downward acceleration, g = 9.8 m/s2, to objects

(If we are at/near Earth’s surface)

This force is felt as the weight of an object. We can approximate g ≅ 10 m/s2,

1 kg × g = 9.8 N ≅ 10N

Thus these approximate conversions hold:

  • 1 kg is about 10 N of force

  • 1/10 of a kg is about 1 N

  • 100 kg is about a kN

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.” There is plenty of gravity from Earth, even at one thousand miles out in space. So they must be weightless for a different reason.

“Ask someone why astronauts in the Space Shuttle or the MIR space station float in their cabins, and quite often you’ll hear that it’s because there’s zero gravity in space…. but that answer is completely wrong. It is true that the pull of gravity is less on the astronauts and their craft, but the pull is only slightly less. With the center of the Earth 3,960 miles away from someone standing on the surface and 4,060 miles away from someone orbiting 100 miles above, the difference in the pull of gravity is only about 5 percent. That means if you weigh 100 pounds on Earth, you would weigh 95 when 100 miles high. No, the real reason that astronauts float around is that they’re in a continuous free fall. “

PBS NOVA Explanation of free fall

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?

The algebraic law for gravity

TBA

Rational functions: Plotting Coulomb’s law or the law of gravity

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

https://www.pbs.org/wgbh/nova/galileo/

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

Barycenters

We have a separate resource on Planetary orbits and barycenters

The Law of universal gravitation and inverse square law

In this section we learn the math equation for Newton’s universal law of gravitation, and the inverse square law.

Related topics

Rational functions: Plotting Coulomb’s law or the law of gravity

Blueberry earth: A thought experiment in planet formation

Gravitational repulsion and the Dipole Repeller

Gravity makes planets spherical: Prove that our world is spherical, not flat

Using science fiction stories to teach physics

Neutron Star – Tides – Physics of a Larry Niven short story

The Integral Trees and The Smoke Ring

Apps

Gizmos Gravity physics apps HTML5

Gravity and orbits

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.