Lecture Notes on Motion, Gravity, and Energy

Three fundamental quantities describe motion: position, velocity, and acceleration.
- Position: How far an object is (in all dimensions) from a reference point.
- Velocity: The rate of change of position (speed is the magnitude of velocity).
- Acceleration: The rate of change of velocity.
- An object can accelerate even when its speed doesn't change if its direction changes.

Newton built upon the work of Galileo to formulate three laws of motion.
- 1st Law: An object moves at a constant velocity (both speed and direction) unless acted on by a force.
- 2nd Law: The acceleration of an object acted on by a force is proportional to the force and inversely proportional to the mass of the object ( $a = F/m$ ).
- 3rd Law: For any force, there is an equal and opposite reaction force.
- These laws govern the motion of all objects in the Universe, except at very high speeds (relativity) and very small scales (quantum mechanics).

Consider a bug flying into the windshield of an oncoming bus.
- Newton's 3rd Law: The forces the bus and bug feel are equal in magnitude but opposite in direction.
- Newton's 2nd Law: The accelerations experienced are different because of the large difference in masses. The bus experiences a tiny acceleration, while the bug experiences a huge acceleration.
 - $a{bus} = F{bug-bus} / M_{bus}$
 - $a{bug} = F{bus-bug} / m_{bug}$

An object in circular motion has a constantly changing velocity, even if its speed is constant, because its direction is changing.
Newton’s 2nd Law implies that there must be a force causing this acceleration.
For a ball on a string, the inward force of the string keeps the ball from flying away.
If the string breaks, the ball will fly away in a straight line.

An object in orbit feels the force of gravity from the central object.
Analogy: Running off a platform on a tall tower.
- The faster you start, the further from the base of the tower you would land.
- With enough initial velocity (about 8 km/s near Earth’s surface), you could fall around the Earth, i.e., orbit.
All objects in orbit stay in their orbital path due to the force of gravity.

Newton understood that gravity causes objects to fall to Earth and holds the Moon in its orbit.
The force acts on both objects equally but in opposite directions (Newton’s 3rd Law).
Gravity is an attractive force between any two objects with mass.
- It depends on the objects’ masses.
- It depends on the distance between them.

Equation for the force of gravity:
- $F{grav} = G \frac{m1 m_2}{r^2}$
 - $G$ is the universal gravitational constant.
 - $m1$ and $m2$ are the two masses.
More mass implies more force.
The distance between the objects is $r$ .
Gravity is governed by an inverse square law.
Inverse square law:
- $F = \frac{G m1 m2}{d^2}$

Gravity has basic properties that set it apart from other forces:
- It acts on mass only (not size nor charge).
- It always supplies an attractive force between any two pieces of matter in the Universe (both attractive & repulsive for FE).
- It is long-ranged and can act over cosmological distances.

$G$ is a very small number, meaning that the force of gravity is negligible unless a very large mass is involved (such as the Earth).
- $G = 6.673 \times 10^{-11} N \cdot m^2/kg^2$
Formula:
- $F = \frac{G m1 m2}{d^2}$

For an extended object, it behaves as if all of its mass were concentrated at its center.

Mass is an intrinsic property of an object – how much of it is there? (measured in kg).
Weight is the force experienced by an object due to gravity (measured in lbs or Newtons).
Weight, unlike mass, depends on the situation.
In a stationary elevator, the weight on the scale is the same as it would be standing on the ground.
If the elevator moves up or down at a constant velocity, the weight on the scale is unchanged.
If the elevator accelerates up, the weight on the scale is higher (you feel heavier).
If the elevator accelerates down, the weight on the scale is lower (you feel lighter).
If the cable is cut, and the elevator falls freely, you feel no weight at all (weightlessness).

Energy is the central unifying theme of all science.
Energy comes in many forms:
- Kinetic energy (energy of motion), including the motion of atoms (temperature) – higher speed means more kinetic energy.
- Radiative energy (energy of light).
- Potential (stored) energy, including mass, which is a form of stored energy ( $E = mc^2$ ).
Energy can change from one form to another, but it is always conserved in total.

When a ball is thrown into the air, it starts with kinetic energy.
When it reaches the top of its motion, it momentarily stops.
The kinetic energy is converted into gravitational potential energy.
The energy is “stored” in the interaction between the ball and Earth.
The energy is almost completely recovered as kinetic energy when the ball falls back to the ground.
High = large GPE; Low = small GPE

Kepler’s 2nd Law: In a given time, a line connecting the Sun to the planet will sweep out an area that is the same in all parts of the orbit.
Thus, the planet moves faster when it is closer to the Sun.
When the planet is closer to the Sun, its GPE is lower, and its KE (and speed) will be higher.
When the planet is further from the Sun, its GPE is higher, and its KE (and speed) will be lower.

Differences between Newton’s and Kepler’s version of planetary orbits:
1. The planets do not technically orbit the Sun; they orbit the center of mass of the system.
  - The center of mass of the Solar System is near the edge of the Sun, so the Sun moves very little, but it does wobble a bit about the center of mass.
2. All objects with mass will orbit, e.g., binary stars.

Newton derived Kepler’s rules from his law of gravity.
Physical laws explain Kepler’s empirical results.
Distant planets orbit more slowly; the harmonic law and the Law of Equal Areas result.
Newton’s Laws were tested by Kepler’s observations.
$P^2 = \frac{4 \pi^2}{GM} r^3$