Today: (Ch. 6 & Review)  Work and Energy  Review (Chapter 1 to 5)

Kinetic Energy The force in the work equation can be found from Newton’s Second Law –W = F Δx = m a Δx The acceleration can be expressed in terms of velocities – Combining: W = ½ m v f ² - ½ m v i ² The quantity ½ m v² is called the kinetic energy –It is the energy due to the motion of the object

Work and Kinetic Energy The kinetic energy of an object can be changed by doing work on the object W = ΔKE –This is called the Work-Energy theorem The units of work and energy are the same –Joules, J –Another useful unit of energy is the calorie 1 cal = J

Work and Force Suppose the person lifts his end of the rope through a distance L The pulley will move through a distance of L/2 W on crate = (2T)(L/2) = TL W on rope = TL Work done on the rope is equal to the work done on the crate

Potential Energy The work done by the gravitational force is always equal to mgh and is independent of the path taken W = mgh An object near the Earth’s surface has a potential energy (PE) : depends on the object’s height, h The potential energy is related to the work done by the force in moving from position 1  2

Potential Energy, final Relation between work and potential energy –ΔPE = PE f – PE i = - W Since W is a scalar, potential energy is also a scalar The potential energy of an object when it is at a height y is PE = m g y –Applies only to objects near the Earth’s surface Potential energy is stored energy –The energy can be recovered by letting the object fall back down to its initial height, gaining kinetic energy

Conservative Forces Conservative forces are forces that are associated with a potential energy function Potential energy can be associated with forces other than gravity The forces can be used to store energy as potential energy Forces that do not have potential energy functions associated with them are called nonconservative forces Potential energy is a result of the force(s) that act on an object Since the forces come from the interaction between two objects, PE is a property of the objects (the system) Potential energy is energy that an object or system has by virtue of its position Potential energy is stored energy It can be converted to kinetic energy

Adding Potential Energy to the Work-Energy Theorem In the work-energy theorem (W = ΔKE), W is the work done by all the forces acting on the object of interest Some of those forces can be associated with a potential energy Assume all the work is done by gravity –Could be any single conservative force –W = - ΔPE = ΔKE KE i + PE i = KE f + PE f Applies to all situations in which all the forces are conservative forces

Mechanical Energy The sum of the potential and kinetic energies is called the mechanical energy Since the sum of the mechanical energy at the initial location is equal to the sum of the mechanical energy at the final location, the mechanical energy is conserved Conservation of Mechanical Energy –KE i + PE i = KE f + PE f –The results apply when many forces are involved as long as they are all conservative forces A very powerful tool for understanding, analyzing, and predicting motion

Conservation of Energy, Example The snowboarder is sliding down a frictionless hill Gravity and the normal forces are the only forces acting on the board –The normal is perpendicular to the object and so does not work on the boarder

Conservation of Energy, Example, ctd. The only force that does work is gravity and it is a conservative force thus Conservation of Mechanical Energy can be applied Let the initial point be the top of the hill and the final point be the bottom of the hill – KE i + PE i = KE f + PE f → ½ m v i ² + m g y i = ½ m v f ² + m g y f With the origin at the bottom of the hill, y i = 0 Solve for the unknown – In this case, v f = ? – The final velocity depends on the height of the hill, not the angle

Charting the Energy A convenient way of illustrating conservation of energy is with a bar chart The kinetic and potential energies of the snowboarder are shown The sum of the energies is the same at the start and end The potential energy at the top of the hill is transformed into kinetic energy at the bottom of the hill

Problem Solving Strategy Recognize the principle –Find the object or system whose mechanical energy is conserved Sketch the problem –Show the initial and final states of the object –Also include a coordinate system with an origin Needed to measure the potential energy Identify the relationships –Find expressions for the initial and final kinetic and potential energies One or more of these may contain unknown quantities

Problem Solving Strategy, cont. Solve – Equate the initial mechanical energy to the final mechanical energy –Solve for the unknown quantities Check –Consider what the answer means –Check that the answer makes sense

Units, Vectors and Significant figures

Forces : Newton's Three Laws & Balancing and Resolving Forces in Components

Velocity and Acceleration & Kinematics Equations

Weight and Apparent Weight & Motion with Friction

Free Fall

Equilibrium & Incline

Projectile Motion

Uniform Circular Motion Centripetal Force and Acceleration

Car on Banked Road Horizontal and Vertical Circular Motion

Newton’s Gravitation Law Orbital Speed and Time Period

Tomorrow: (First Exam)  Exam in the class