Gravity

Review Question

What are Kepler’s laws of planetary motion?

Review Question

What time of year does the Earth move the fastest in its orbit around the Sun?

Review Question

What does it mean for something such as the Moon to be in synchronous rotation?

Discussion

Although the Moon is in synchronous rotation, we can actually see 59% of its surface from the Earth. Explain why this is the case using Kepler’s laws.

Review Question

What are Newton’s 3 laws of motion?

Discussion

Is there a force acting on the Moon? How can you tell?

What was Newton thinking?

Consider tossing a baseball. It travels a certain distance before it hits the ground.

Now image throwing it as hard as you can. It travels farther before it hits the ground.

Now imagine throwing it even harder.

The Moon is falling

The nearly circular orbit of the Moon is constantly accelerating toward the Earth.

The Moon is constantly falling toward the Earth.

Discussion

If I swing a ball in a circle over my head with a short string and a long string with the same speed, which ball has the greater force acting on it? Explain why.

The higher the speed an object moves in a circle the greater acceleration and the force needed to hold it in that circle.

The force on the planets closest to the Sun has to be greater than that on the planets farther away.

Force on the planets depends on distance

Newton’s Universal Law of Gravity

• Every mass attracts every other mass through a force called gravity

• The force is directly proportional to the product of their masses

• The force is inversely proportional to the square of the distance between them

221

d

MMGF

Discussion

Consider the gravitational force between two objects with mass M1 and M2 separated by a distance d. How would the gravitational force change if the distance between them increases to 3 d. How will it change in the distance in decreased to 0.1 d?

Why the square of the distance?

An inverse square central force law is required to get orbits that are conic sections, i.e. orbits that are elliptical.

Why is the force of gravity proportional to the mass?

All objects, regardless of their mass, fall with the same acceleration. Because F = ma,

m

Fa To keep the acceleration

constant, the force must vary proportional to the mass.

Discussion

Newton’s third law tells us that the force of the Sun on the Earth is the same as the force of the Earth on the Sun. Why then does the Earth orbit the Sun instead of the other way around?

d

The distance to use is the distance between the two spheres centers.

Gravitational forces between spherical masses

Discussion

Suppose a new planet is discovered out in the Kuiper belt. This planet has twice the mass of the Earth but is also twice the radius. Is the surface gravity of this new planet greater than, less than or the same as the surface gravity of the Earth?

Discussion

You dig a very deep mine shaft. As you get closer to the center of the Earth, does your weight increase or decrease? Why?

(Hint: consider what the force of gravity will be at the very center of the Earth.)

Escape Velocity

If an object is thrown up with a high enough velocity it will leave Earth forever. For Earth this velocity is about 11 km/sec.

If the Earth were the same mass but twice the radius, would the escape velocity be greater or less than 11 km/sec? Why?

Discussion

The escape velocity from the Moon is less than that from the Earth because of the Moon's

a) lower density. b) smaller mass. c) smaller radius. d) distance from the Earth.

Newton’s law of gravity explained the ocean tides

Because the force of gravity falls off rapidly with distance, the pull of the Moon and Sun are stronger on the side of Earth that is closer to, or faces, the Moon or Sun.

And it is weakest farthest from the Moon or Sun.

Spring Tides and Neap Tides

The Sun also contributes to the Earth’s tides. When the Sun and Moon line up to produce higher tides, this is called spring tides. Neap tides occur when the Moon and Sun Partially cancel each other.

What phases of the Moon do spring and neap tides occur?

Note: Tidal forces

Tidal forces are not a new kind of force that stretch and squeeze things.

Are the name given to the difference in accelerations across an finite sized object due to differences in distance from a gravitating source

Discussion

Due to tidal friction, what happens to the rotation rate of the Earth over time?

The Earth-Moon system

The Moon exerts a force on the tidal bulges on Earth, trying to twist the tidal bugles back to face the Moon. This torque slows the rotation of the Earth.

Conservation of angular momentum

Any object that is spinning or orbiting has angular momentum which is equal to the mass × velocity × radius.

In the absence of an external torque, or twisting force, the angular momentum will remain constant.

Discussion

The Moon is slowing the Earth’s rotation and the Earth is losing angular momentum. Where does that angular momentum go?

Orbit of the Moon

Due to conservation of angular momentum, the slowing of Earth’s rotation requires the Moon to move faster in its orbit. Because the force of Earth’s gravity does not change, this means the Moon is getting farther away, by about 2 inches per year.

Discussion

What do you think will be the ultimate fate of the Earth Moon system?

Fate of the Earth Moon System

Over billions of years the length of the month and the length of the day will be the same, about 47 days. The Moon will stationary in the sky and will be too small for total solar eclipses.