PHY2053 Lecture 11 Conservation of Energy. Conservation of Energy Kinetic Energy Gravitational Potential Energy

Transcription

1 PHY2053 Lecture 11 Conservation of Energy Conservation of Energy Kinetic Energy Gravitational Potential Energy

2 Symmetries in Physics Symmetry - fundamental / descriptive property of the Universe itself [ vacuum ] Colloquial: Symmetric Laws of Physics are the same at any point in space [ translational invariance ] Conservation of Momentum [Ch 7] Laws of Physics are the same at any point in time [ time invariance ] Conservation of Energy [today s lecture] Physics term: Parity 2

3 More practical aspect there are different, mathematically equivalent ways to formulate Newton s laws all these calculations predict certain quantities will be conserved for a closed system (0 net external force) energy, momentum, angular momentum.. existence of conserved quantities simplifies otherwise complicated calculations Key concepts: learn to recognize and exploit conserved quantities conserved quantities derived from Newton s laws solutions immediately satisfy Newton s laws 3

4 Energy Conservation term closed system means: no net external force is acting upon any element of the system The total energy of a closed system does not change over time: total energy before = total energy after textbook implies that the Universe is a closed system The total energy in the Universe is unchanged by any physical process next: define change of energy (work), energy itself 4

5 Concept of Work colloquial meaning of work: effort which produces a result. analogy in terms of mechanics: Effort Force, F Result Displacement r interested in displacement due to force F θ r W = F r cos( ) angular term cos(θ) projects force displacement SI unit: Joule [ J ]; relation to calorie: 1 cal = 4.2 J 5

6 Work: signed scalar quantity Work can be positive, negative, and zero depending on the orientation of the force to the displacement F θ θ = 90 r F r F θ r θ < 90 cosθ > 0 W > 0 θ = 90 cosθ = 0 W = 0 θ > 90 cosθ < 0 W < 0 6

7 Total Work in a Closed System start with total work on a particular object i W i = i F i r cos( i )= recall the definition of a closed system i F i =0 r i vector sum, has to be zero in all directions i F i cos( i ) F i cos( i )=0 i W i = r i F i cos( i ) =0 7

8 Kinetic Energy, Definition consider impact of work on the velocity of an! object start from 1D motion, works in all three (x, y, z) W = F x x = ma x x a x x = v2 f,x W = ma x x = m v2 f,x v 2 i,x 2! 2 = m v2 f,x 2 v 2 i,x! 2 m v2 i,x 2 K = m v2 2 v 2 W = K f K i = K Work Energy Theorem 8

9 Example #1: Mass Driver A mass driver is a device which uses magnetic fields to accelerate a container (mass). Predicted commercial uses include launching people and cargo to bases on the Moon. The common way to specify mass drivers is to quote the kinetic energy that an object will have when leaving the driver, if it started from rest. For a 1 MJ mass driver, compute the muzzle velocity of a) a 0.5 kg projectile b) a 50 kg projectile 9

10 Mass driver notes pt 1 10

11 Mass driver notes pt 2 11

12 Gravitational Potential Energy Near Earth near Earth, the usual orientation of coordinate systems is so that the positive y axis points up the force of gravity has only one component, in the y-direction: Fy = mg only y displacement, y matters for computing work: W = FG,y y = mg y consider a vertical shot upwards, vf = 0 W = K = Kf Ki = 0 ½mvf 2, also = mg y gravity did negative work, removing kinetic energy 12

13 Energy Conservation Law where did the kinetic energy go? temporarily stored in gravitational field define potential energy Ugrav = Wgrav = mg y computes how much kinetic energy could be released if we let gravity work across y work-energy theorem: W = K; K W = 0 K + U = 0 ( K + U ) = 0 sum of kinetic and potential energy does not change define E = K + U, then E is constant in time 13

14 Choice of Zero Point, Near Earth Due to conservation of energy, only changes in potential energy are really relevant for kinematics The absolute value of potential energy at a point in space is arbitrary - up to an additive constant We have the freedom to pick a convenient point in space and declare that the potential energy at that point equals 0 J All other potential energies are then computed relative to that point, based on U = U(y) U(0) U(y) = U + U(0) = mg y + 0 = mg (y 0) 14

15 Example #1: Rollercoaster A roller-coaster is barely moving as it starts down a ramp of height h. The first figure it encounters is a loop of radius R. How high must the ramp be so that the roller-coaster never loses contact with the rails? h R 15

16 Rollercoaster notes pt 1 16

17 Rollercoaster notes pt 2 Comment: Given that the total height of the loop is 2R, this is not really much taller than the loop itself. The ratio of the height of the ramp and the height of the loop is 2.5R / 2R = the ramp has to be only 25% taller than the loop for the rollercoaster to clear the highest point in the loop and stay in contact with the rails. 17

18 More Realistic: Dissipative (Non-conservative) Forces friction converts mechanical energy into heat heat does not store mechanical energy therefore, there is no point in defining a heat or frictional potential energy friction always opposes motion, so Wfriction < 0 extend the law of energy conservation to account for non-conservative forces: (Ki + Ui) + WNC = (Kf + Uf) 18

19 Gravitational Potential Energy, Planetary Scales derivation requires math beyond baseline calculus U grav = G m 1 m 2 for gravitational potential at planetary scales, there already exists a usual convention: potential energy infinitely far away from a planet is = 0 convention: an object with positive total energy can escape a planet (will not fall back to the planet) allows easy computation of escape velocities for objects starting from any R from the planet s center r 19

20 Example #2: Hyperbolic Comet A comet not bound to the Sun will only pass by the Sun once. It will trace a hyperbolic trajectory through the Solar system. Compute the minimum velocity of a hyperbolic comet when it is roughly 1 A.U. away from the Sun. The mass of the Sun is MS = kg. 1 Astronomical Unit is the distance from the Earth to the Sun, 150 million km. Does the velocity depend on the mass of the comet? 20

21 Hyperbolic Comet notes 21

22 Next Lecture: Hooke s Law, Elastic Potential Energy Power