# Unification of the laws of the Earth and the Universe Why do planets appear to wander slowly across the sky?

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1 October 19, 2015 Unification of the laws of the Earth and the Universe Why do planets appear to wander slowly across the sky? Key Words Newton s Laws of motion, and Newton s law of universal gravitation: force, mass, gravity, weight Unless otherwise noted, copied pictures are taken from wikipedia.org

2 October 19, 2015 Contents n Founder of Mechanics (review) and Isaac Newton n Newton s laws of motion n Momentum conservation law n The universal law of gravitation

3 Last week n Introduction: - Predictability of our physical world - Stonehenge n The birth of modern Astronomy - Historical background: AD 2 nd C ~ 16C - Observations:16~17C n The birth of Mechanics - Galileo Galilei - The founder of experimental science

4 The birth of Mechanics Birth of Mechanics n Mechanics: the branch of science that deals with the motions of material objects such as - a rock rolling down a hill - a ball thrown into the air - a sailboat skimming over the waves - planets orbiting around the Sun n Until 17 th C, why things move the way they do was not understood

5 The birth of Mechanics n Galileo Galilei ( ): The first person to record observations of the heavens with telescope

6 The birth of Mechanics Galileo Galilei n To understand Galileo s study moving objects, ultimately the workings of the solar system, we have to begin with precise definitions of three terms: speed, velocity, and acceleration

7 The birth of Mechanics Mechanics: speed, velocity, acceleration n Definitions of speed, velocity, and acceleration Speed: distance divided by the time it takes to travel that distance Ex) 60 km/hr (scalar) Velocity: the same numerical value as speed, but including direction Ex) 60 km/hr due west (vector) Acceleration: a measure of the rate of change of velocity. (vector) Whenever an object changes speed or direction, it accelerates

8 The birth of Mechanics The founder of experimental science n Galileo devised an ingenious experiment to determine the relationships among distance, time, velocity, and acceleration. Designed purely to measure acceleration, with an inclined plane crafted of brass and hard wood, by increasing the angle of elevation of the plane 30 o 20 o 10 o

9 The birth of Mechanics The founder of experimental science n Galileo found that any accelerating toward Earth s surface, no matter how heavy or light, falls with exactly the same constant acceleration in free falling (no air resistance). A 1,000kg B 1kg Free falling Surface of Earth Acceleration_A = a_a = Acceleration_B = a_b = g

10 n Experiment from Leaning Tower of Pisa, probably never performed??

11 The birth of Mechanics Galileo s work in speed, velocity, acceleration n In general, the velocity of an accelerating object that starts from rest is proportional to the length of time that it has been falling Velocity = constant a x time t Acceleration n Galileo s work also demonstrated that the distance covered by an accelerating object depends on the square of the travel time. Distance traveled = ½ x acceleration x time 2

12 And yet it moves! By Galileo?? after being forced to recant his belief that the earth moves around the sun, in inquisition.

13 With Galileo s work, scientists began to isolate and observe the motions of objects in nature, and summarize their results into mathematical relationships.

14 Introduction - Isaac Newton n Isaac Newton ( ): synthesized the work of Galileo and others in a statement of the basic principles that govern the motion of everything in the Universe

15 Introduction - Isaac Newton n Isaac Newton ( ): synthesized the work of Galileo and others in a statement of the basic principles that govern the motion of everything in the Universe è Newton s laws of motion: simple and obvious statements to represent results of centuries of experiments and observations

16 October 19, 2015 Contents n Founder of Mechanics (review) and Isaac Newton n Newton s laws of motion - 1 st law - 2 nd law - 3 rd law n Momentum conservation law - (Linear) Momentum - Angular momentum n The universal law of gravitation - Gravity - Weight and gravity

17 Newton s laws of motion

18 Newton s laws of motion The 1 st Law n Newton recognized two different motions Observations: - An object is in uniform motion if it travels in a straight line at constant speed - All other motions are called acceleration involving changes of speed, changed of direction, or both

19 Newton s laws of motion The 1 st Law: law of inertia In physics n Inertia: the tendency of a body to maintain its state of rest or uniform motion unless acted on by an external force. A body at rest tends to stay at rest because of its inertia, while a moving body tends to keep moving because of its inertia. n Force is defined as something that produces a change in the state of motion of an object

20 Newton s laws of motion The 1 st Law: law of inertia n The First Law A moving object will continue moving in a straight line at a constant speed, and a stationary object will remain at rest, unless acted on by an unbalanced force. Define the concept of force as something to cause a mass to accelerate

21 Newton s laws of motion The 1 st Law: example inertia Bus was moving to the right, and stopped suddenly.

22 Newton s laws of motion The 2 nd Law n Observations: It is easier to lift a child than an adult. Force = mass (kg) x acceleration ( m/s 2 ) : F=ma n If we know the forces acting on a system of known mass, we can predict its future motion Stationary or changed à An object acceleration is a balance between two factors: force and mass

23 Newton s laws of motion The 2 nd Law: law of acceleration n The Second Law The acceleration produced on a body by a force is proportional to the magnitude of the force and inversely proportional to the mass of the object. It tells us the exact magnitude of the force necessary to cause a given mass to achieve a given acceleration

24 Newton s laws of motion The 2 nd Law: example F 1 = m 1 a 1 F 2 = m 2 a 2 F 1 = F 2 à m 1 < m 2, a 1 > a 2 where m1, m2, and a1, a2 are the masses and accelerations of a golf ball and a truck, respectively Picture modified from

25 Newton s laws of motion The 2 nd Law: unit of force n F = ma Force = mass [kg] x acceleration [m/s 2 ] n Unit of force: newton [N] = [kg*m/s 2 ] (newton) (kg-meter-per-second-squared)

26 Newton s laws of motion The 3 rd Law n Forces always act simultaneously in pairs Action ç è Re-action Ex) When you hold a book up, the book feels heavy in your hands as it presses down, but your hands exert an equal and opposite force on the book

27 Newton s laws of motion The 3 rd Law: law of action-reaction n The Third Law For every action there is an equal and opposite reaction. Whenever a force is applied to an object, that object simultaneously exerts an equal and opposite force

28 Newton s laws of motion The 3 rd Law: example Force propels rocket in opposite direction Force produced from engine

29 Newton s laws of motion Newton s Laws at Work n Newton s three laws of motion form a comprehensive description of all possible motions Ex) A rocket launching to the space

30 Newton s laws of motion Newton s Laws at Work n Newton s three laws of motion form a comprehensive description of all possible motions Ex) A rocket launching to the space n However, they do not say anything about the nature of forces. In fact, much of the progress of science has been associated with the discovery and elucidation of the forces of nature

31 Momentum conservation law

32 Momentum conservation Momentum speed mass n Tendency of a moving object to remain in motion depends on the mass of the object and on its speed

33 Momentum conservation Momentum speed mass n Tendency of a moving object to remain in motion depends on the mass of the object and on its speed n Observation: the higher the mass and the higher the speed, the more difficult it is to stop the object or change its direction of motion è (Linear) Momentum = mass (kg) x velocity (m/s): p = mv

34 Momentum conservation Conservation of Momentum n Newton s 2 nd law imply: If no external forces act on a system, the changes in the total momentum of a system is zero è law of conservation of linear momentum P = mv, F = ma, v = at Change of (P) = change of (mv) = ma X change of (t) F=0 è change of (P) = 0 * When a quantity does not change, the quantity is conserved.

35 Conservation of momentum: example Collision of two billiard balls Collision before after m 1 v 1 m 2 m 1 m 2 v 2 first ball momentum=m 1 v 1 second ball momentum=0 at rest first ball momentum=0 at rest second ball momentum=m 2 v 2 Total momentum of the system before collision m1v1 + m2v2 = m1v1 Total momentum of the system after collision m1v1 + m2v2 = m2v2

36 Conservation of momentum: example Collision before after m 1 v 1 m 2 m 1 m 2 v 2 first ball momentum=m 1 v 1 second ball momentum=0 at rest first ball momentum=0 at rest second ball momentum=m 2 v 2 When there is no external force, for example the friction force, m1v1 = m2v2 because the total momentum is always conserved.

37 Momentum conservation Angular Momentum n Angular Momentum - Just as on object keeps moving in a straight line unless an external force acts on it, an object that is rotating will keep rotating unless a twisting force acts to make it stop. - A twisting force: torque

38 Momentum conservation Angular Momentum n Angular Momentum (cont d) - A twisting force: torque (τ ) Ex) A spinning top will keep spinning until the friction between its point of contact and the floor slows it down - Tendency to keep rotating is angular momentum :angular momentum : torque

39 Momentum conservation Angular Momentum n Two factors increase angular momentum - The rate of spin: the faster an object spins, the harder it is to stop - The distribution of mass: object with more mass, or with mass located farther away from the central axis of rotation, have greater angular momentum

40 Momentum conservation Angular Momentum n Two factors increase angular momentum - The rate of spin: the faster an object spins, the harder it is to stop - The distribution of mass: object with more mass, or with mass located farther away from the central axis of rotation, have greater angular momentum Ex) A solid metal and an air-filled tire of same diameter and rate of spin have different angular momentum

41 Universal law of gravitation

42 The universal law of gravitation Gravity n Gravity: the most obvious force in our daily lives - It holds us down and keeps us from floating off into space - It guarantees that when you drop things they fall down n * Gravity s quantitative properties were studied by Galileo and his contemporaries * Newton revealed its universality

43 The universal law of gravitation Gravity n Newton s apple?? - Newton s question: Can the gravity that causes an apple to move downward extend far outward to the Moon, supplying the force that keeps it from flying off?

44 The universal law of gravitation Gravity n Newton s apple?? - Newton s hypothesis: there must be a force acting on the Moon going around Earth, not moving in a straight. This force is the same force that made the apple fall.

45 The universal law of gravitation Gravity n Newton s apple?? (cont d) - He realized that the orbits of all the planets also could be understood if gravity was as force found throughout the universe

46 The universal law of gravitation Newton s law of universal gravitation n Newton s law of universal gravitation: Between any two objects in the universe there is an attractive force (gravity) that is proportional to the masses of the objects and inversely proportional to the square of the distance between them m1 R Gravity Force = G m1 m2 R m2 m1: mass1 m2: mass2 R: distance G: constant

47 The universal law of gravitation Newton s law of universal gravitation n Newton s law of universal gravitation: units force of gravity = [ G x mass1 x mass2 ] / [distance] 2 Force of gravity is in newtons [N], masses are in [kg], distance is in [meter], and G is a number called the gravitational constant big G è The more massive objects are, the greater the force between them will be; the farther apart they are, the less the force will be

48 The universal force of gravity Weight and Gravity n Weight and Gravity - Law of universal gravitation says that there is a force between any two objects in the universe - Gravitational attraction between you and Earth pull down and the ground exerts a force equal and opposite to that of gravity When you are standing on a scale, the scale shows the exerted opposing gravity è your weight! (weight = gravity) - Weight depends on where you are - Mass is an absolute value: the amount of matter

49 Next topic is, Energy: Chapter 2

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