Potential Energy. Serway 7.6, 7.7;

Transcription

1 Potential Energy Conservative and non-conservative forces Gravitational and elastic potential energy Mechanical Energy Serway 7.6, 7.7; Practice problems: Serway chapter 7, problems 41, 43 chapter 8, problems 5, 11, 53, 59, 61 - Lecture 22

2 Potential energy -another approach to calculating the work done -only possible for conservative forces -leads to the idea of energy as a conserved quantity The basic idea: Work done by a force is equal to the decrease in potential energy: Work = (change in potential energy)

3 Example: Gravitational Work mg To lift the block to a height y requires work (by F P :) F P = mg W P = mgy y When the block is lowered, gravity does work: mg W g1 = mg.s 1 = mgy or W g2 = mg.s 2 = mgy y s 1 s 2

4 Work done (against gravity) to lift the box is stored as gravitational potential energy U g : U g =(weight) x (height) = mgy (uniform g) When the block moves, (work by gravity) = P.E. lost W g = U g The position where U g = 0 is arbitrary. U g is a function of position only. (It depends only on the relative positions of the earth and the block.) The work W g depends only on the initial and final heights, not on the path.

5 Conservative Forces A force is called conservative if the work done (in going from A to B) is the same for all paths from A to B. A path 1 B path 2 W 1 = W 2 An equivalent definition: For a conservative force, the work done on any closed path is zero. Total work is zero.

6 Concept Quiz The diagram at right shows a force which varies with position. Is this a conservative force? a) Yes. b) No. c) We can t really tell.

7 For every conservative force, we can define a potential energy function U so that W A B = U = U A U B Examples: gravity (uniform g) : U g = mgy, where y is height gravity (exact, for two particles, a distance r apart): U g = GMm/r, where M and m are the masses Ideal spring: U s = ½ ks 2, where s is the stretch electrostatic forces (we ll do this in January)

8 Quiz An charged particle moving on the x axis has electrostatic potential energy 10 J at x=0, and 15 J at x=2 m. The work done by the electrostatic force on the charge when the charge is moved from the origin to x=2m is A) 5 J B) 10 J C) 15 J D) -5 J E) -10 J

9 Examples of non-conservative forces: friction drag forces in fluids (e.g., air resistance) Friction forces are always opposite to the relative velocity (the direction of f changes as v changes). Work done to overcome friction is not stored as potential energy, but converted to thermal energy.

10 Conservation of mechanical energy If only conservative forces do work, potential energy is converted into kinetic energy or vice versa, leaving the total constant. Define the mechanical energy E as the sum of kinetic and potential energy: E K + U = K + U g + U s +... Conservative forces only: W = U Work-energy theorem: W = K So, K+ U = 0; which means E is constant in time

11 Example: Pendulum The pendulum is released from rest with the string horizontal. a) Find the speed at the lowest point (in terms of the length L of the string). v f b) Find the tension in the string at the lowest point, in terms of the weight mg of the ball.

12 Quiz: Pendulum A student who doesn t like energy methods would prefer to use his favourite kinematical formula, v 2 -v 02 =2ad But what can you say about the acceleration as the pendulum falls? v f A) the average acceleration is equal to g B) the average tangential acceleration is equal to g C) the average vertical acceleration is equal to g D) the average horizontal acceleration is equal to g E) None of the above is true - Lecture 22

13 Example: Block and spring. v 0 A block of mass m = 2.0 kg slides at speed v 0 = 3.0 m/s along a frictionless table towards a spring of stiffness k = 450 N/m. How far will the spring compress before the block stops?

14 Question From where should the pendulum be released from rest, so that the string hits the peg and stops with the string horizontal? θ L/2 L/2