Unit 5

Gravitation

• Newton’s Law of Universal Gravitation

• Kepler’s Laws of Planetary Motion

Into to Gravity Phet Simulation

Today: Make sure to collect all data.Finished lab due tomorrow!!

Universal Law of Gravitation

Developed by Isaac Newton

All objects attract each other through gravitational force.

Depends on the masses of the objects and the distance between them• Gravity increases when mass increases

• Gravity decreases when distance increases

Both objects will feel the same force

Universal Law of Gravitation: Relationships

•Gravity and distance: Inversed squared

distance

Universal Law of Gravitation: Relationships

•Gravity and mass: Direct

The gravitational force of attraction between two objects would be increased by:

A. Doubling the mass of both objects only

B. Doubling the distance between the objects only

C. Doubling the mass of both objects and doubling the distance between the objects

D. Doubling the mass of one object and doubling the distance between the objects

Answer: A

Law of Gravitation Practice

• The gravitational force between two objects is 50N. If the distance between the objects is cut in half, what happens to the force?

200 N (quadruples)

• The gravitational force between two objects is 50N. If the mass of each object is tripled, what happens to the force?

450 N (50 x3 x3)

Law of Gravitation Practice

Law of Gravitation Practice

Which graph best represents the gravitational force F that a rocket experiences as it

travels a distance D away from the surface of the Earth?

**Note that RA represents the edge of the atmosphere**

Answer: BEarth’s gravitational reach far

exceeds the end of the atmosphere! Think about it, the moon is still

influenced by Earth’s gravity and it is far past our atmosphere!

Law of Gravitation Practice

A 72 kg woman stands on top of a very tall ladder so she is one Earth radius above Earth's surface (double the distance she normally is)

Step 1: Figure out her normal weight on the surface of the Earth:

Step 2: Gravity and distance have an inverse squared relationship so if she is double the distance away the force of gravity is 1/4 th as strong. So divide her normal weight by 4.

Gravity and the Distance from Center of the Mass

• Gravity is greater the closer you get to the center of the mass

Law of Gravitation Practice

A rocket moves away from the surface of Earth. As the distance r from the

center of the planet increases, what happens to the force of gravity on the rocket?

A. The force increases directly proportional to r

B .The force increases directly proportional to 𝑟2

C. The force doesn’t change

D. The force becomes zero after the rocket loses the contact with Earth

E. The force decreases inversely proportional to 1𝑟2

Universal Law of Gravitation Calculations

• The relationship between gravity, mass and distance can be calculated using the below equation:

Where:

F = gravitational force mB = mass of second object

G = gravitational constant mA = mass of first object

d = distance between the two objects

Law of Gravitation PracticeTwo spherical objects have masses of 200 kg and 500 kg. Their centers are separated by a distance of 25 m. Find the gravitational attraction between them.

Law of Gravitation PracticeTwo spherical objects have masses of 3.1 x 105 kg and 6.5 x 103 kg. The gravitational attraction between them is 65 N. How far apart are their centers?

Law of Gravitation PracticeA 1 kg object is located at a distance of 6.4 x106 m from the center of a larger object whose mass is 6.0 x 1024 kg.

• What is the size of the force acting on the smaller object? 9.8N

• What is the size of the force acting on the larger object?

Newton’s Third Law – the forces are equal so the answer is 9.8N

Law of Gravitation Formula Rearrangement

If we know the gravitational force between two objects that have the same mass, how can we rearrange our equation to solve for those masses?

• 𝑛𝑒𝑒𝑑 𝑡𝑜 𝑠𝑜𝑙𝑣𝑒 f𝑜𝑟 𝑡ℎ𝑒 𝑚𝑎𝑠𝑠𝑒𝑠:

𝑚a x 𝑚b = 𝑚2

𝐹 = 𝐺𝑚2

Fd2 = Gm2

Fd2 = m2

𝑚 = 𝐹d2

G

d2

G

Relationships that affect Gravity

•More mass = greater gravity

• Same mass in smaller radius = greater gravity

Why do you think that is?

What is gravity?

•What do you think gravity is?

•What do you think causes gravity?

•Is gravity a push/pull force?

Gravity: From Newton to Einstein

Demo: General Relativity

•Why do more dense things create a stronger gravity?• They create a deeper bend in the fabric of space-time

• “Space tells masses how to move and masses tell space how to curve/bend”

General Relativity• Gravity is not a pulling force, but instead a warping of space-time

which creates “dimples” that objects “fall” into

• Too much mass in too small an area, the warping causes a black hole!

Calculating the gravity caused by a mass

• Use the formula below to calculate the gravity created by any mass:

**where G is the same gravitational constant used before, m is the

mass of the object and r is the radius of the object**

Calculating Gravity Practice

• Jupiter’s mass is hundred’s of times that of the Earth. Why is the acceleration due to gravity on that planet only 3x that of Earth? • Jupiter is much bigger/less dense

Compute g at a distance of 4.5 x 107m from the center of a spherical object whose mass is 3.0 x 1023 kg.

Calculating Gravity Practice

Kepler’s Laws of Planetary Motion

• 1st Law: Law of Ellipses

• 2nd Law: Law of Equal Areas

• 3rd Law: Law of Periods

Kepler’s 1st Law

• The orbits of the planets are elliptical (not circular) with the Sun at one focus of the ellipse.

Kepler’s 2nd Law• Law of equal areas

• Planets move faster closer to the sun, slower further away, but over the same period of time, the area of the circle will be equal

Keplers 3rd Law

• Shows the relationship between the distance of planets from the Sun, and their orbital periods.

• Further away = longer time it takes to make one orbit around the sun

Keplers 3rd Law

• The Law of Periods: The square of the period of any planet is proportional to the cube of the planets distance from the sun.

•T3 r2

T = Period of the object• Period is the time it takes for the object to orbit the sun

• Measured in years

r = distance to the sun• Measured in au

(astronomical units=distance from Earth to Sun=1.50 x 1011 m)

Keplers 3rd Law• If you plot the values for objects in the solar system you get the below graph

• Straight slope indicates that the ratio of T3 = r2 is the same for all planetary bodies in the solar system

• Means that you can mathematically compare two bodies:

Kepler’s 3rd Law PracticeGiven Earth has a period of 1 year and a radius from the sun of 1 au, if Mars has a period of 1.88 Earth years, how far is Mars from the sun?