Determine the value of x so that v w = O. where

Chapter 7 – Energy of a System 7.1 Systems and Environments system—valid systems may: be a single object or particle be a collection of objects or particles be a region of space vary with time in size and shape system boundary—an imaginary surface that divides the Universe into the system and the environment environment—surrounds the system 7.2 Work Done by a Constant Force work W done on a system by a constant force equals the force times the displacement times cos θ θ is between the force and the direction of displacement (think components) the displacement is from the point of application of the force no work is done if the force does not move through a displacement or if the force is perpendicular to the direction of motion the SI unit of work is the newton-meter¸ or joule (J) work is an energy transfer: positive work transfers energy to the system, negative work transfers energy from the system 7.3 The Scalar Product of Two Vectors A∙B=ABcosθ scalar product: W=F ∆rcosθ=F∙∆r ´ ´ ´ ^ ^ ^ ´ ^ ^ ^ if A=A i+x j+Ayk z , and B=B i+x j+Byk z , ´ ´ then A∙B=A B +x Bx+A y y z z 7.4 Work Done by a Varying Force xf For a force that varies with position, W= F ∫x x that is, the area under the xi curve of force vs position (along x-axis) Spring Force (Hooke’s Law): Fs=−kx k is the spring constant, and the units are N/m 1 2 W s k x 2 7.5 Kinetic Energy and the Work-Kinetic Energy Theorem 1 2 KE= m v 2 1 2 1 2 W totKE= mv −2m v f 2 i work-kinetic energy theorem: When work is done on a system and the only change is the speed of the system, the net work is equal to the change in kinetic energy of the system (see above equation) 7.6 Potential Energy of a System potential energy is determined by the configuration of the system (i.e. position) gravitational potential energy: U gmgh 1 2 elastic potential energy: U s 2 k x 7.7 Conservative and Nonconservative Forces internal energy includes the energy associated with the temperature of a system conservative forces have two properties: The work done by a conservative force on a particle depends only on initial and final positions of the particle, not the path taken The work done on the particle if it moves through a closed path is zero (ends at the beginning) gravitational force is a conservative force Work done by nonconservative forces depends on the path taken mechanical energy is the sum of the kinetic and potential energies of a system 7.8 Relationship Between Conservative Forces and Potential Energy potential energy function U: when only conservative forces work on the system, such that the work done within the system is the negative of the change in the potential energy −dU Fx= dx 7.9 Energy Diagrams and Equilibrium of a System stable equilibrium: any movement away from here results in a force directed back to this position unstable equilibrium: any movement away results in a force directed away from this position neutral equilibrium: when U is constant over some region. Chapter 8 – Conservation of Energy 8.1 Analysis Model: Nonisolated System (Energy) In nonisolated systems, energy crosses the boundary of the system from interacting with the environment. In isolated systems, the system does not interact with its environment. Work is a method of transferring energy to a system by applying force over a displacement Mechanical waves transfer energy by propagating a disturbance through air or another medium Heat is a method of energy transfer driven by temperature difference between system and environment Matter transfer is when matter physically crosses the boundary of the system Electrical transmission involves energy transfer through electric currents Electromagnetic radiation cross the boundary of the system to transfer energy If the total energy of a system changes, it is because some has crossed the boundary by a method like those listed above. Energy is never created or destroyed; it is always conserved ∆ Esystem T where T is the amount of energy transferred 8.2 Analysis Model: Isolated System (Energy) In an isolated system: ∆ Emech∆ K+∆U=0 8.3 Situations Involving Kinetic Friction W − f d=∆K ∑ otherk ∆ E = f d internak change in internal energy due to constant friction force ∑ W other∆K+∆ E internal 8.4 Changes in Mechanical Energy for Nonconservative Forces ∫¿=0 if a nonconservative force acts within an isolated s∆K+∆U+∆ E ¿ ∫ ¿ if nonconservative forces act on nonisolated system:er=¿∆K+∆U+∆E ¿ ∑ ¿ 8.5 Power dE instantaneous power P: P= dt or the time rate of energy transfer W average power: P= ∆t the SI unit of power is the watt (W) and equals 1 J/s U.S. customary unit is the horsepower (hp): 1 hp = 746 W a unit of energy, the kilowatt-hour (kWh) is equal to 3.60 × 10 J