#### Transcript Work, Energy and Power

Work, Energy and Power Ms Houts AP Physics C Chapters 7 & 8 ©2008 by W.H. Freeman and Company Definition of Work The work done by a constant force F applied to an object that moves through a displacement Dx is defined as W F cos Dx Fx Dx Work is a scalar quantity that can be positive or negative. ◦ Forces applied in the direction of motion is positive. ◦ Forces applied opposite the direction of motion fk ©2008 by W.H. Freeman and Company Definition of Kinetic Energy An object moving with velocity v and mass m has a kinetic energy given by: KE mv 1 2 2 ©2008 by W.H. Freeman and Company Work-Kinetic Energy Theorem The total work done on a particle is equal to the change in its kinetic energy. Wtotal DKE mv mv 1 2 2 f 1 2 2 i Work-Kinetic Energy Example 1 A 6-kg box is raised from rest a distance of 3 meter by a vertical force of 80 N. Find the work done by the force, the work done by gravity, and the final speed of the box. Graphical Interpretation of Work Work is the area under the Force-position graph. Work with a variable force Work with a variable force The work done by a force F that varies with position x is given by W x2 F dx x x1 This integral equals the area under the Fversus-x curve. Work-Kinetic Energy Example 2 A spring that obeys Hooke’s Law rests on a frictionless surface. ◦ Hooke’s Law gives the force of a spring as F=-kx Find the work done by the spring force when it is stretched from x = 0 to x = xf. Practice Exercises with Work Further practice- pages 160-162 Work in three dimensions Only the component of the force in the direction of the displacement does work. W F cos Dx Fx Dx If a force and displacement have components F Fx ˆi Fy ˆj Fz kˆ and Ds Δxˆi Dyˆj Dzkˆ then the work done by the force is given by W Fx Dx Fy Dy Fz Dz Dot Product The dot product is the product of two vectors A and B, where we consider only the part of A that lies in the direction of B, or the part of B that lies in the direction of A. A B AB cos where is the angle between A and B. A B Ax Bx Ay B y Az Bz Work as the Dot Product The work done by a constant force F over a displacement ∆s is W F Ds If the force varies with position, then the work is given by s2 W F ds s1 Properties of Scalar Products Power Power is the rate at which work is done. DW dW P or in derivative form P Dt dt Since W Fx Dx DW Dx P Fx Fv Dt Dt or in derivative form dx P Fx dt Power in 3 dimensions A force F acting on an object moving with velo city v supplies a power of P Fv Practice problems, p. 162 Conservative Forces A force is conservative if the total work it does on a particle is zero when the particle moves along any closed path returning to its initial position. Potential Energy Functions Doing work against a conservative force stores energy. When the conservative force does work on a particle, that energy is released. Let U be the potential energy associated with a conservati ve force. Let W be the work done by that conservati ve force. W DU x2 DU W Fx dx if the force varies with position. s2 x1 DU F ds s1 ©2008 by W.H. Freeman and Company Gravitation Potential Energy The work done by the gravitational force in lifting up a weight a distance y is negative, since the force is opposite the displacement. Fg mgˆj Ds yj W mgy The change in gravitational potential energy is positive. DU W (mgy) mgy Spring Potential Energy s2 x1 s1 0 DU F ds Fx dx x1 DU (kx)dx 0 Using the Potential Energy to Find the Force Since the potential energy is the negative of the integral of the force function: x2 DU U f U i Fx dx x1 Then the negative of the derivative of the potential energy function is the force function. dU Fx dx Equilbrium A particle is in equilibrium if the net force acting on it is zero. Since the force is the derivative of the potential energy function, the equilibrium points can be found graphically from a potential energy graph by finding places where the slope of the graph is zero. In stable equilibrium, a small displacement results in a restoring force That accelerates the particle back toward its equilibrium position. ©2008 by W.H. Freeman and Company In unstable equilibrium, a small displacement results in a force that accelerates the particle away from its equilibrium position. ©2008 by W.H. Freeman and Company In neutral equilibrium, a small displacement results in zero force and the particle remains in equilibrium. ©2008 by W.H. Freeman and Company