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Chapter 8 Potential Energy and Conservation of Energy 8-6 Conservation of Energy Wext = ΔE 8-1 Work and Potential Energy 8-2 Path Independence of Conservative Forces 8-3 Determining Potential Energy Values 8-4 Conservation of Mechanical Energy 8-5 Work Done on a System by an External Force and Thermal Energy due to Friction Kariapper Ch9-page 1 8-6 Conservation of Energy Energy of a System Work Kinetic Energy Theorem For a Single Object: W = ΔK Conservation of Energy Principle For a system of objects W = ΔE Change in Kinetic energy Work done by an (external) force on the object/system Change in Kinetic energy Change in Energy ΔE = ΔK + ΔU ΔEth + ΔEint Change in Change in Potential energy thermal energy Change in internal energy However we will not worry about DEint in phys101, and we’ll deal only with situation where DEint =0 Kariapper Ch9-page 2 8-6 Conservation of Energy Energy of a System A particle-earth system A spring-block system Fext F g Fg is not an external force Fext Fs Fs is not an external force If we choose our system to be a particle or a block only, then Fg and Fs will be an external force to our system (which is a single body now) Kariapper Ch9-page 3 8-2 Work and Potential Energy Gravitational Potential Energy A particle-earth system F F g g initial W = ΔK + ...?... final If Fext = 0 for this system then W = 0, but we know that DK is not zero as the ball falls.. So where is the energy coming from to increase its kinetic energy? We associate what is called a Potential Energy Ug with the configuration of the earth particle system, the change of which is the negative of the work done by gravitational force: DUg = -Wg Kariapper Ch9-page 4 8-3 Path Independence of Conservative Forces Conservative Forces Work done by a conservative force does not depend on the path between the initial and final point. W1 W1=W2 W2 In another words: The net work done by a conservative force on a particle moving around any closed path is zero. W=0 Conservative Force Gravitational Force Frictional Force Spring Force Kariapper Non-Conservative Force Drag force Ch9-page 5 8-3 Path Independence of Conservative Forces Non-Conservative Forces Non-Conservative Force Work depends on the path Example: Kariapper Ch9-page 6 8-3 Determining Potential Energy Values Definition of Potential Energy It is defined for a system of two or more objects It is defined only for conservative forces xf DU c W c F dx Gravitational Potential Energy, Ug xi yf DU g Fg dy W g mg Dy yi Elastic Potential Energy, Us xf 1 2 2 DU s Fs dx W s k x f x i 2 xi Kariapper Ch9-page 7 8-3 Determining Potential Energy Values Checkpoint Checkpoint: A particle moves from x =0 to x=1, under the influence of a conservative force as shown. Rank according to DU, most positive first (descending order). Kariapper Ch9-page 8 8-3 Determining Potential Energy Values Example What is the U of the sloth-earth system if our reference point is • at the ground • at a balcony floor that is 3.0 m above the ground • at the limb • 1 meter above the limb? Kariapper Ch9-page 9 8-4 Conservation of Mechanical Energy Mechanical Energy Emech = K+U Mechanical energy Kinetic energy Potential energy If there is no external acting on a system (W = 0) and its thermal energy is constant (no non-conservative forces such as friction, DEth = 0), then its mechanical energy is conserved. 0 = DK+DU OR Kariapper Mechanical Energy is conserved Kf+Uf= Ki+Ui=constant Ch9-page 10 8-4 Conservation of Mechanical Energy Checkpoint Rank according to speed at point B. Ball sliding on a Frictionless Ramp Kariapper Ch9-page 11 8-4 Work Done on a System No friction involved We have a block-floor-earth system. The applied force F is external to the our system. The law of conservation of energy states that the work done by external force on a system is equal to the change of energy W = DE 0 same height Fd= DK+DU Kariapper Ch9-page 12 8-4 Work Done on a System friction involved We have a block-floor-earth system, this time there is friction. The friction force is now an internal force. However as a result of the friction force, there will be a loss in the kinetic energy which appear as an increase in the thermal energy (converted to heat) of the system. W = DE W = DEmech+DEth 0 F d= DK+DU+DEth It can be shown that the change in thermal energy is equal to the negative of the work done by the friction force: DEth= fk d = - Wf Therefore the conservation of energy principle can also be stated in the form: W + Wf = DEmech fk d Kariapper Ch9-page 13 8-4 Work Done on a System Example By what distance d is the spring compressed when the block stops? W = DEmech+DEth Kariapper OR W + Wf = DEmech DEth = - Wf = fk d Ch9-page 14