Physics 1D03 - Lecture 22 Potential Energy Serway and Jewett 8.1 – 8.3 Work and potential energy Conservative and non-conservative forces Gravitational and elastic potential energy Conservation of Mechanical Energy

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

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.

Physics 1D03 - Lecture 22 Conservative Forces A force is called “conservative” if the work done (in going from some point 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. Later you’ll see this written as: Total work is zero. path 1 path 2 A B W 1 = W 2

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?

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

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.

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 E is constant in time (ie: dE/dt=0) Conservation of mechanical energy

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). b)Find the tension in the string at the lowest point, in terms of the weight mg of the ball. L

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

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?

Physics 1D03 - Lecture 22 Example From where (what angle or height) should the pendulum be released from rest, so that the string hits the peg (located at L/2) and stops with the string horizontal?  L/2

Physics 1D03 - Lecture 22 Example m1m1 m2m2 Two masses are connected over a pulley of Radius R and moment of inertia I. The system is released from rest and m1 falls through a distance of h. Find the linear speeds of the masses.

Physics 1D03 - Lecture 22 Solution

Physics 1D03 - Lecture 22 Summary Conservative and non-conservative forces Potential energy : work = P.E. lost Gravitational and elastic (spring) P.E. Mechanical energy in conservative systems.