Basic Mechanics

Units

Velocity and Acceleration Speed: Time rate of change of position. Velocity: Speed in a specific direction. Velocity is specified by a Vector. Acceleration: The time rate of change of velocity (speed and/or direction). Acceleration is also a vector.

Gravitational acceleration of the Earth = 9.8 m/s 2 10 m/s 2

Constant velocity  Zero acceleration Constant acceleration in the same direction as v  Increasing velocity Constant acceleration opposite of v  Decreasing velocity

Velocity: the average velocity over an infinitesimal (very short) time interval Acceleration: the average acceleration over an infinitesimal (very short) time interval

Instantaneous Velocity Instantaneous velocity is slope of tangent at any point

1D Kinematics Equations for Constant Acceleration

 x = v 0 t + 1/2 at 2 (parabolic) v = v 0 + at (linear) v 2 - v 0 2 = 2a  x

Momentum and Angular Momentum Momentum ( 운동량 ) is the combination of both mass and velocity. Mathematically: p = mv Momentum is a vector, and it is always conserved (can be neither created nor lost, only transferred). Angular Momentum ( 각운동량 ) is momentum of spinning/rotating or revolving. It is also a vector and is also conserved. Mathematically: L = mv  r

mm v v m 2m v 0.5v m 0.5m v 2v

Momentum and Angular Momentum Momentum ( 운동량 ) is the combination of both mass and velocity. Mathematically: p = mv Momentum is a vector, and it always conserved (can be neither created nor lost, only transferred). Angular Momentum ( 각운동량 ) is momentum of spinning/rotating or revolving. It is also a vector and is also conserved. Mathematically: L = mv  r = mvr sin 

Angular momentum: m(vxr)=mvr sin 

Newton's Laws of Motion ( 뉴턴의 운동 법칙 ) 1.If the sum of the forces on an object is zero, then the object's velocity will remain constant. Since neither speed nor direction of motion changes that we always have motion in a straight line when there are no forces. 2.Acceleration experienced by an object is directly proportional to the amount of force exerted on it. 3.Action and Reaction: When two bodies interact, they create equal and opposite forces on each other. F = ma

V: constant First law

Second law

Third law Action and Reaction

The Newton's law tells us that if there are no forces acting on an object then its motion is in a straight line. So how do we produce circular motion? We have to apply a force. The force constantly change the direction of of the object's motion without altering the speed (not velocity!) of the object. If we swing the object around in a circle above our heads, we can feel the tension ( 장력 ) as we swing the object. If we were to suddenly cut the string the object would continue off on a tangential ( 접선 ) line from the spot it was released.

Universal Law of Gravity One day Newton was out sitting under an apple tree looking at the Moon. When he saw an apple fall, he wondered if the force that was keeping the Moon in orbit around the Earth was the same force that made the apple fall to the ground. It would mean that the Moon is always falling toward Earth in the way we described above. He also then considered that it must be gravity that holds planets in orbit about the Sun.

 If the Moon is falling a little towards the Earth, just like an apple dropped on the surface, why does the Moon travel around the Earth in an orbit instead of falling onto it?  The way to answer this question is to consider what would happen if there was no gravity acting:  Q: How far would the Moon travel in a straight line in 1 sec if there were no gravity acting?  A: About 1000 meters.  At the same time, the Moon's motion along this straight-line path would also cause it to move away from the Earth.  Q: How far away from the Earth would the Moon move in 1 second if no gravity were acting?  A: About m  In round numbers, the amount the Moon falls towards the Earth due to gravity is just enough to offset the straight-line path it would take if gravity were not acting to deflect it. This balance effectively closes the loop.  We have therefore reached a conclusion:  The Moon is really falling around the Earth!

Universal Law of Gravity He studied Kepler's law of planetary motion and his own laws of motion to come up with a Universal Law of Gravitation. Given two masses, M 1 and M 2 the force between them is given by where r denotes the distance between two masses, and G (=6.67x m 3 kg -1 s -2 ) means a gravitational constant.

Example (1) The Fall of an Apple Stand on the Earth and drop an apple. What is the force of the Earth on the apple? F = GM earth M apple / R earth 2 What is the apple's acceleration (Newton's 2nd Law of Motion): a apple = F / M apple = GM earth / R earth 2 = 6.7   (6.0  ) / (6.4  10 7 ) 2 = 9.8 m/s 2 Note that the mass of the apple (M apple ) had divided out of the equation. This means that the acceleration due to gravity is independent of the mass of the apple.

R Earth M Earth M apple

Example (2) Newton's 3rd Law of Motion states that all forces come in equal yet opposite pairs. What force does the the apple apply in return upon the Earth? F = GM earth M apple / R earth 2 How much does the Earth accelerate towards the apple? a earth = F / M earth = GM apple / R earth 2 This can be rewritten to give the acceleration of the Earth in terms of the acceleration of the apple towards the Earth as a earth = a apple x (M apple /M earth ) where a apple =9.8 m/s 2, and the ratio of the Mass of the apple to the Mass of the Earth is very small number. For a typical 200g apple, this works out to be about m/s 2.

Kepler's Three Law 1.The orbit of every planet is an ellipse with the sun at one of the foci. 2.A line joining a planet and the sun sweeps out equal areas during equal intervals of time as the planet travels along its orbit. This means that the planet travels faster while close to the sun and slows down when it is far from the sun. 3.The squares of the orbital periods of planets are proportional to the cubes of the semi-major axes (the "half-length" of the ellipse) of their orbits. a 3 =K T 2 a

ellipse

DiameterMass Semi-major axis (AU) Rotational period Orbital period (yr) Sun- Terrestrial planets Mercury Venus Earth11111 Mars Gas planets Jupiter Saturn Uranus Neptune Dwarf planets Ceres Pluto

a 3 =K T 2

Energy Kinetic Energy ( 운동 에너지 ): the energy of motion for any object with mass, m, and velocity, v, its kinetic energy, K E, is given by Potential Energy ( 위치 에너지 ): the amount of work that can potentially be done. For an object, with mass, m, under the gravitational attraction of another with mass, M, and a distance, r, away the potential energy, P E, is Total Mechanical Energy = K E +P E

mm v v m 2m v 0.5v m 2m v 2/3v 1/3v

칠석 ( 七夕 )

Vega Altair