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## MOTION AND GRAVITY

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MOTION AND GRAVITY

- The key LAWS of the key branch of physics known as MECHANICS were formulated by Isaac Newton
- Three LAWS of MOTION
- The LAW of GRAVITY
- The LAWS of CONSERVATION OF ENERGY and MOMENTUM give a more general way to understand motion
- While a physics course would spend the whole first semester on these laws we’ll just get a taste of them!

Speed, Velocity and Acceleration

- Speed = distance traveled per time (car at 110 km/hr or 70 mph)
- Velocity is a speed + a direction (70 mph NE)
- Acceleration is a change in velocity per time: speed and/or direction (10 km/s2)

- Isaac NEWTON (1642-1727) of Woolsthorpe, England is the most important scientist in history.
- His work completely changed the way educated people looked at the world. Effectively, Newton was the founder of PHYSICS as well as THEORETICAL ASTRONOMY
- HE CO-INVENTED THE CALCULUS (w/ Leibnitz)
- HE DID PIONEERING WORK IN
- OPTICS: PRISM, REFLECTING TELESCOPE
- MECHANICS AND GRAVITY: his Philosophiae Naturalis Principia Mathematica, (pub. 1684) reflected work he'd mostly done in 1665-1666.
- Newton was knighted, and became first president of the Royal Society, later Director of the Mint.

NEWTON'S LAWS OF MOTION

- 1. An object at rest remains at rest and an object moving at a certain velocity retains that velocity unless a FORCE acts on it.
- Aristotle's view: forces were needed merely to keep something moving at a constant speed
- Newton realized friction or air resistance were forces that slowed things down
- Galileo had already understood this.

Which of the following is true?

- A. You can have acceleration not equal zero, but velocity equal to zero
- B. You can have acceleration equal to zero, but velocity not equal to zero
- C. You can accelerate without changing your speed
- D. A and B.
- E. A, B and C.

Which of the following is true?

- A. You can have acceleration not equal zero, but velocity equal to zero
- B You can have acceleration equal to zero, but velocity not equal to zero
- C. You can accelerate without changing your speed
- D. A and B.
- E. A, B, and C.

Newton’s SECOND LAW

- The core of Newtonian mechanics, it allows trajectories of cannon balls, rockets, planets, comets, stars and galaxies to be computed.
- F = m a
- is the most important relation in physics; one can equivalently write
- a = F/m
- This clearly says less massive objects obtain larger accelerations from the same force.
- Think of stepping on the gas and going from 0 mph to 60 mph in 10 seconds: your acceleration is 6 mph/s (forwards)
- 2nd Law Applet

Momentum and Force

- Momentum = mass X velocity (p = mv)
- It takes a force to change a body’s momentum
- Slightly more general version of Newton’s 2nd Law: F = p/t
- Think of a 1000 kg car colliding with a 6000 kg truck head on -- if they have the same speeds the truck has 6 times the momentum and will push the car down the road

More on the 2nd Law

- Breaking takes you from 60 mph back to 0 in 4 sec
- or a negative acceleration of 15 mph/s.
- These are VECTOR equations -- with magnitude and direction
- Velocity = distance covered / time
- V = d/t
- Acceleration = change in velocity/time change
- a= V/t
- - both the Speed and Direction are needed
- I.e. 50 mph to the East is the same speed, but different velocity, from 50 mph to the North
- Going around a curve at a constant speed DOES involve an acceleration (you feel pushed to one side of the car, right?)

Newton’s Third Law

- 3. EVERY ACTION (FORCE) HAS AN EQUAL AND OPPOSITE REACTION.
- Forces don't act in isolation:
- the Earth pulls the Moon and the Moon pulls back on the Earth;
- we push down and back on the ground with our muscles, it pushes us up and forward;
- a rower or gondolier pushes water (or canal bottom) in one direction and the scull or gondola goes the other way;
- a rocket expels gases rearward and it flies forward.

Changing an object’s momentum requires

- A. Gravity
- B. Applying a force
- C. Applying a torque
- D. Friction
- E. None of the above

Changing an object’s momentum requires

- A. Gravity
- B. Applying a force
- C. Applying a torque
- D. Friction
- E. None of the above

Newton’s second law, F = m·a, (force = mass x acceleration), means that with no force,

- A. Objects remain at rest
- B. An object’s speed doesn’t change
- C. An object’s velocity doesn’t change
- D. B and C.

Newton’s second law, F = m·a, (force = mass x acceleration), means that with no force,

- A. Objects remain at rest
- B. An object’s speed doesn’t change
- C. An object’s velocity doesn’t change
- D. B and C

Conservation Laws in Astronomy

- Momentum
- Angular Momentum
- Energy

Conservation of (linear)

Momentum is implied by

Newton’s Laws of Motion.

One ball hits another, exerts a force, which accelerates

Second ball (2nd law); 3rd Law says opposite force decelerates the first ball

Angular Momentum Conservation

- AM = m x v x r (mass x velocity x distance)
- Orbital AM conservation says no push needed to keep Earth orbiting and also faster motion at perihelion than aphelion: v x r = constant
- Rotational AM conservation says Earth keeps spinning on its axis and also faster spin when contracted: ballerina, gas cloud making planets

Conservation of Energy

- Energy comes in many forms but three classes can contain them all:
- Kinetic (energy of motion)
- Radiative (energy of light or electromagnetic radiation)
- Potential (stored energy -- gravitational, chemical, atomic, mass-energy)

Thermal or Heat Energy

- Random kinetic energy of atoms and molecules
- Heat or thermal energy is the sum total of all of them
- Temperature is related to the average energy

Gravitational Potential and Kinetic Energy

- No KE, maximum gravitational potential energy at top of throw
- Maximum KE, minimum gravitational PE when thrown and when caught
- KE = (1/2)mv2
- Energy of Thrown Ball

Temperature is a measure of:

- A. How much heat an object contains
- B. How fast atoms are moving
- C. How hot you feel when you touch something
- D. Energy

Temperature is a measure of:

- A How much heat an object contains
- B How fast atoms are moving
- C How hot you feel when you touch something
- D Energy

A cake is baking at 400 degrees. If you briefly touch the cake you will not be burned. Touch the metal pan for the same length of time and you will be burned. Why?

- A. The metal is hotter than the cake
- B. The metal is denser than the cake–there are more atoms per unit volume
- C. The metal is a better conductor
- D. B. and C.
- E. All of the above

A cake is baking at 400 degrees. If you briefly touch the cake you will not be burned. Touch the metal pan for the same length of time and you will be burned. Why?

- A. The metal is hotter than the cake
- B. The metal is denser than the cake–there are more atoms per unit volume
- C. The metal is a better conductor
- D. B and C.
- E. All of the above

NEWTON'S LAW OF GRAVITY

- The ATTRACTIVE FORCE OF GRAVITY IS DIRECTLY PROPORTIONAL TO THE PRODUCT OF THE MASSES
- AND INVERSELY PROPORTIONAL TO THE SQUARE OF THE DISTANCE, r, BETWEEN THEM.
- where Newton’s gravitational constant
- G = 6.673 x 10-11 m3 kg-1 s-2

Gravitational Acceleration: 1

- Combine 2nd Law of Motion w/ Law of Gravity
- ACCELERATION DUE TO GRAVITY, g, OF AN OBJECT IS PROPORTIONAL TO ITS MASS AND INVERSELY PROPORTIONAL TO THE SQUARE OF THE DISTANCE FROM ITS CENTER.

Gravitational Acceleration: 2

- Example: if mE = ME and r = RE

g = 9.80 m s-2 (or 32 ft/s2)

YOU SHOULD VERIFY THIS CALCULATION!

Gravitational Acceleration

Newton’s law of gravity is F = G m1 m2 / d2Can this be used to find the force between the Sun and a planet? If so, what is d?

- A. No
- B. Yes, d is the diameter of the Sun
- C. Yes, d is the diameter of the planet
- D. Yes, d is the distance from the Sun to the planet

Newton’s law of gravity is F = G m1 m2 / d2Can this be used to find the force between the Sun and a planet? If so, what is d?

- A. No
- B. Yes, d is the diameter of the Sun
- C. Yes, d is the diameter of the planet
- D. Yes, d is the distance from the Sun to the planet

When I drive my car at 30 miles per hour, it has more kinetic energy than it does at 10 miles per hour.

- Yes, it has three times as much kinetic energy.
- Yes, it has nine times as much kinetic energy.
- No, it has the same kinetic energy.
- No, it has three times less kinetic energy.
- No, it has nine times less kinetic energy.

When I drive my car at 30 miles per hour, it has more kinetic energy than it does at 10 miles per hour.

- Yes, it has three times as much kinetic energy.
- Yes, it has nine times as much kinetic energy.
- No, it has the same kinetic energy.
- No, it has three times less kinetic energy.
- No, it has nine times less kinetic energy.

Weight v. Mass

- Weight (Newtons, dynes) is the force due to gravity acting on a mass (amount of matter, kilograms, grams) so
- W = m g (special case of F = m a). Since gravity gets weaker a greater distances, you actually weigh less at the top of a building than you do at its base, even though your mass hasn't changed.
- Since Atlanta is about 300 m above sea level, you weigh a little less here than in Savannah
- -- at sea level, and closer to the center of the earth.
- You weigh more in an elevator as it just accelerates to go up and less in one that accelerates to go down;
- you are weightless in one that is falling w/o support!

Weight and Weightlessness

- Take a scale in an elevator with you. No cable free fall
- Fast leap from a tower constant free-fall (weightlessness)

Figuring Out the Law of Gravity: 1

- Newton compared the acceleration the Moon feels compared to that felt at the surface of the Earth.
- Knew the Earth-Moon distance was about 60 x RE
- Found the inertial ("centripetal") acceleration, a, due to rotation at speed v and at distance r (experiment: rock swung on string)

Figuring Out the Law of Gravity: 2

Start from vM = circumference of orbit divided by period

- This gives, aM = 2.7 x 10-3 m s-2
- Newton realized: aM = g/(3625) = g/(rEM /RE )2
- concluded the INVERSE SQUARE RELATION OF GRAVITY ON DISTANCE was LIKELY to be true EVERYWHERE.

Gravity Keeps the Moon from Flying off on a Tangent: “Constantly Falling”

Circular Velocity and Escape Velocity

- Newton also showed that the general shape of a BOUND ORBIT was an ELLIPSE (with a circle as a special situation)
- and that the general shape of an ESCAPE ORBIT was a HYPERBOLA (with a parabola as a special case).
- The simplest case: a CIRCULAR orbit, just skimming the earth

Orbit Shape Depends on Speed

- v = vc : circular orbit
- vc < v < vesc : elliptical orbit w/ center of E at near focus;
- Both BOUND (NEGATIVE ENERGY ORBITS)
- v = vesc: parabolic escape orbit: reaches infinity with no energy left (ZERO ENERGY ORBIT)
- v > vesc
- hyperbolic escape orbit: reaches infinity still moving away (POSITIVE ENERGY ORBIT)
- For earth, vesc = 11.2 km/s, or about 25,000 mph!

Newton DERIVED Kepler’s Laws

- FUNDAMENTAL LAWS explain EMPIRICAL ONES
- Consider a general circular orbit of a low mass object around a much more massive one:

- This is Kepler's Third Law!
- Newton also derived Kepler’s first and second laws, but these are actually harder (you do this in a sophomore, not freshman, physics course).

Weighing Astronomical Bodies

- For example, to get the mass of the Sun + Earth (basically just Sun)

m = 2.0 x 1030 kg

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