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Potential Energy • • • Work and potential energy Conservative and non-conservative forces Gravitational and elastic potential energy Physics 1D03 - Lecture 22 mg Gravitational Work To lift the block to a height y requires work (by FP :) FP = mg WP = FPy = mgy When the block is lowered, gravity does work: y mg Wg1 = mg.s1 = mgy or, taking a different route: y Wg2 = mg.s2 = mgy s1 s2 Physics 1D03 - Lecture 22 Work done (against gravity) to lift the box is “stored” as gravitational potential energy Ug: Ug =(weight) x (height) = mgy (uniform g) When the block moves, (work by gravity) = P.E. lost Wg = -DUg • The position where Ug = 0 is arbitrary. • Ug is a function of position only. (It depends only on the relative positions of the earth and the block.) • The work Wg depends only on the initial and final heights, NOT on the path. 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 45o above the horizontal. What is its speed as it hits the ground. Physics 1D03 - Lecture 22 Conservative Forces path 1 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. B A path 2 W1 = W2 An equivalent definition: For a conservative force, the work done on any closed path is zero. Total work is zero. Physics 1D03 - Lecture 22 Concept Quiz The diagram at right shows a force which varies with position. Is this a conservative force? a) b) c) d) Yes. No. We can’t really tell. Maybe, maybe not. Physics 1D03 - Lecture 22 For every conservative force, we can define a potential energy function U so that WAB = -DU = UA -UB Note the negative Examples: Gravity (uniform g) : Ug = mgy, where y is height Gravity (exact, for two particles, a distance r apart): Ug = - GMm/r, where M and m are the masses Ideal spring: Us = ½ kx2, where x is the stretch Electrostatic forces (we’ll do this in January) 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 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 + Ug + Us + ... Conservative forces only: W = -DU Work-energy theorem: W = DK So, DK+DU = 0; which means that E does not change with time: dE/dt = 0 Physics 1D03 - Lecture 22 Example: Pendulum L 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). vf Physics 1D03 - Lecture 22 Example: Pendulum 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 vf Physics 1D03 - Lecture 22 Example: Block and spring v0 A block of mass m = 2.0 kg slides at speed v0 = 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