Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity. Copyright 2009 Pearson Education, Inc.

Transcription

1 Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity

2 How do we describe motion? Precise definitions to describe motion: Speed: Rate at which object moves speed = distance time Example: speed of 10 m/s! # " units of m s Velocity: Speed and direction Example: 10 m/s, due east Velocity is a vector! Acceleration: Any change in velocity; units of speed/time (m/s 2 ) Acceleration is a vector! $ & %

3 Acceleration of Gravity All falling objects accelerate at the same rate (not counting friction of air resistance). On Earth, g 10 m/s 2 : speed increases 10 m/ s with each second of falling.

4 Clicker Question You throw an object straight up. When the object is at its highest point, a) its velocity and acceleration are zero. b) its velocity is zero and its acceleration is directed downward. c) its velocity is zero and its acceleration is directed upward. d) its velocity and acceleration are both nonzero.

5 Acceleration of Gravity (g) Galileo showed that g is the same for all falling objects, regardless of their mass. Feather and Hammer Drop Apollo 15 demonstration

6 Circular Motion A car is driving around in a circle of radius R with a constant speed v. In which direction in the car s acceleration? 1. The acceleration is zero. 2. The acceleration is directed towards the center of the circle. 3. The acceleration is directed radially away from the center of the circle. 4. None of the above

7 Centripetal (radial) Acceleration For circular motion, the object is always accelerating. a rad = v2 r v = 2πr T (for constant speed only!)

8 Isaac Newton Invented laws of motion Invented Law of Gravity Applicable not only on Earth, but for celestial objects as well! Co-invented Calculus Much more: Experiments with light; first reflecting telescope, Sir Isaac Newton ( )

9 Newton s Laws of Motion What causes an acceleration? Forces are pushes or pulls an object experiences due to an interaction with another object. An object can t exert a force on itself Forces are vectors Net force exerted on object causes an acceleration: F net = F i = ma i Unit of force is a Newton (N). 1 N = 1 kg m/s 2

10 Common Forces Gravity: Weight is force due to gravity F =w= mg 1 lb = 4.45 N Normal Force (contact force) Friction (static and kinetic) Tension Drag (air resistance) Electromagnetic force

11 Clicker Question A person is moving upwards in an elevator at a constant speed. If the person is standing on a scale, the reading of the scale (which shows the upward normal force on the person) is 1. showing the person s normal weight. 2. less than the person s normal weight. 3. greater than the person s normal weight.

12 Newton s 3 rd Law All forces that an object experiences come from interactions with other objects. For two interacting objects, the forces experience due to the other are equal in magnitude and opposite in direction

13 Gravity Near the Earth s surface the gravitational force is given by F = mg, but is this force independent of distance? Newton postulated that gravity is also responsible for planetary and lunar orbits. The Moon s orbit requires an acceleration a Moon = m/s 2 1 = g 3600 Moon s orbital radius is d=60 R earth. Thus acceleration decreases as the inverse square. a F 1 d 2

14 Newton s Law of Gravity Force must also depend on both mass AND be consistent with Newton s 3 rd law: F g m 1m 2 r 2 F g = Gm 1m 2 r 2 G = 6.67 x N m 2 kg -2 This applies to point-like objects or spherical objects (for spheres, r is distance between centers)

15 Weight Near Earth The weight of a body of mass m is the force of gravity it experiences. For an object on the surface of the Earth, For an object above the surface of the Earth, its weight is given by w = Gm m r 2 w = F g = Gm m R 2 = mg

16 Clicker Question Suppose you stood on a planet having a radius twice as long as the Earth s radius and a mass 2 times more massive than the Earth. How much would you weigh on this planet? a) The same as on Earth b) You would be weightless c) Two times more than on Earth d) Two times less than on Earth e) Four times less than on Earth

17 When at rest on the launching pad, the force of gravity on the space shuttle is quite huge the weight of the shuttle. When in orbit, some 200 km above Earth s surface, the force of gravity on the shuttle is 1. nearly as much. 2. about half as much. 3. nearly zero (micro-gravity). 4. zero. (Neglect changes in the weight of the fuel carried by the shuttle.)

18

19 If the Sun suddenly collapsed to become a black hole, the Earth would 1. leave the Solar System in a straight-line path. 2. spiral into the black hole. 3. continue to circle in its usual orbit.

20 Newton s Law of Motion Explain Kepler s Laws Combination of F=ma and Newton s Law of Gravity yield trajectory For bound orbits, trajectory is an ellipse For unbound orbit, trajectory is a hyperbola

21 Orbital Motion What is the velocity for a circular orbit? F = ma G Mm r 2 = mv2 r GM = v 2 r GM v = r Trajectories of mass for different initial speeds

22 Kepler s 3 rd Law Revisited For circular orbits (with orbiting object s mass very small compared to other mass): Period = circumference/velocity P = 2πr P = v orb 2πr GM/r For planets going around Sun, P (years) = r(au) 3/2 P 2 =4π 2 r3 GM

23 Kepler s 3 rd Law Revisited For non-circular ellipses, the more general version (deduced from Newton s equations of motion and gravity) is P 2 =4π 2 a 3 G(M 1 + M 2 ) a is the semi-major axis of the ellipse Mass of Sun is known from measuring period of Earth and distance to Sun. This method is used to measure mass of distant objects (stars, black holes, etc )

24 Conservation of Angular Momentum Angular momentum describes amount of rotation. Torque is required to change angular momentum. Forces in a radial direction exert no torque! Kepler s 2 nd law is explained by angular momentum being conserved

25 What is Energy? Energy Hard to define! Property of the system Related to the amount that something can change the condition of matter. Forces cause some energy of one form to be converted into other forms. Energy of system is increased if external forces do work on the system

26 Basic Types of Energy Kinetic (motion) Radiative (light) Potential Energy Gravitational Electric/Chemical Rest-Mass Energy Thermal Energy Forces cause transfer of energy types K = 1 2 mv2

27 Gravitational Potential Energy On Earth, it depends on an object s mass (m). the strength of the acceleration of gravity (g). the distance an object could potentially fall.

28 Gravitational Potential Energy In space, an object or gas cloud has more gravitational energy when it is spread out than when it contracts. A contracting cloud converts gravitational potential energy to thermal energy. U grav = GMm r

29 Mass-Energy E = mc 2 A small amount of mass can release a great deal of energy. Nuclear fusion can give off energy if atoms lose mass due to fusion. Particles can be created out of kinetic energy (in particle accelerators) or colliding photons!

30 Thermal Energy: The collective kinetic energy of many particles (for example, in a rock, in air, in water) Thermal energy is a measure of the total kinetic energy of all the particles in a substance. It depends on temperature AND amount of material Temperature is related to the average kinetic energy of each atom/molecule

31 Conservation of Energy If all the forms of energy are accounted for, the total energy of a closed system remains constant. Examples Roller coaster (kinetic and grav. potential) Solar panels Gasoline (chemical potential into thermal and kinetic) Radioactive decay (rest-mass into kinetic and/or radiative energy)

32 Clicker Question Three balls are thrown off a cliff with the same speed, but in different directions. Which ball has the greatest speed just before it hits the ground? A. Ball A B. Ball B C. Ball C D. All balls have the same speed

33 Understanding Elliptical Orbits from Energetics More gravitational energy; less kinetic energy Less gravitational energy; more kinetic energy Total orbital energy stays constant. Total orbital energy (gravitational + kinetic) stays constant Potential energy is negative. If total energy is less than zero, bound orbit (elliptical) U grav = GMm r

34 Escape Velocity If an object gains enough orbital energy, it may escape (change from a bound to unbound orbit). Escape velocity from Earth 11 km/s from sea level (about 40,000 km/hr). v esc = 2GM r