1. Physics 1D03 - Lecture 22 Potential Energy Work and potential energy Conservative and non-conservative forces Gravitational and elastic potential energy

2. Physics 1D03 - Lecture 22 Gravitational Work s2s2 y s1s1 mgmg When the block is lowered, gravity does work: W g1 = mg. s 1 = mgy or, taking a different route: W g2 = mg. s 2 = mgy y mgmg F P = mg To lift the block to a height y requires work (by F P :) W P = F P y = mgy

3. Physics 1D03 - Lecture 22 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.

4. Physics 1D03 - Lecture 22 Example A rock of mass 1kg is released from rest from a 10m tall building. What is its speed as it hits the ground ? The same rock is thrown with a velocity of 10m/s at an angle of 45 o above the horizontal. What is its speed as it hits the ground.

5. Physics 1D03 - Lecture 22 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. An equivalent definition: For a conservative force, the work done on any closed path is zero. Total work is zero. path 1 path 2 A B W 1 = W 2

6. Physics 1D03 - Lecture 22 Concept Quiz a)Yes. b)No. c)We can’t really tell. d)Maybe, maybe not. The diagram at right shows a force which varies with position. Is this a conservative force?

7. Physics 1D03 - Lecture 22 For every conservative force, we can define a potential energy function U so that W AB   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 kx 2, where x is the stretch Electrostatic forces (we’ll do this in January) Note the negative

8. Physics 1D03 - Lecture 22 Non-conservative forces: friction drag forces in fluids (e.g., air resistance) Friction forces are always opposite to v (the direction of f changes as v changes). Work done to overcome friction is not stored as potential energy, but converted to thermal energy.

9. Physics 1D03 - Lecture 22 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 that E does not change with time: dE/dt = 0 Conservation of mechanical energy

10. Physics 1D03 - Lecture 22 Example: Pendulum vfvf 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). L

11. Physics 1D03 - Lecture 22 Example: Pendulum vfvf The pendulum is released from rest at an angle θ to the vertical. a)Find the speed at the lowest point (in terms of the length L of the string). θ

12. Physics 1D03 - Lecture 22 Example: Block and spring v0v0 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?