Chapter 5EnergyWorkProvides a link between force and energyThe work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacementWork, cont. F is the magnitude of the force ∆ x is the magnitude of the object’s displacement θ is the angle between x)cosF(W∆θ≡and∆F xrrWork, cont.This gives no information about the time it took for the displacement to occur the velocity or acceleration of the objectWork is a scalar quantityMore About WorkThe work done by a force is zero when the force is perpendicular to the displacement cos 90° = 0If there are multiple forces acting on an object, the total work done is the algebraic sum of the amount of work done by each forceMore About Work, cont.Work can be positive or negative Positive if the force and the displacement are in the same direction Negative if the force and the displacement are in the opposite directionKinetic EnergyEnergy associated with the motion of an objectScalar quantity with the same units as workWork is related to kinetic energy2mv21KE =Work-Kinetic Energy Theorem When work is done by a net force on an object and the only change in the object is its speed, the work done is equal to the change in the object’s kinetic energy Speed will increase if work is positive Speed will decrease if work is negativenet f iW KE KE KE= − = ∆Types of ForcesThere are two general kinds of forces Conservative Work and energy associated with the force can be recovered Nonconservative The forces are generally dissipative and work done against it cannot easily be recoveredConservative Forces A force is conservative if the work it does on an object moving between two points is independent of the path the objects take between the points The work depends only upon the initial and final positions of the object Any conservative force can have a potential energy function associated with itNonconservative ForcesA force is nonconservative if the work it does on an object depends on the path taken by the object between its final and starting points.Examples of nonconservative forces kinetic friction, air drag, propulsive forcesFriction Depends on the Path The blue path is shorter than the red path The work required is less on the blue path than on the red path The work done by friction depends on the path, so friction is a non-conservative forcePotential Energy Potential energy is associated with the position of the object. “Position” makes sense only if the object is part of a system. A system is a collection of objects interacting via forces or processes that are internal to the system Potential energy is really a property of the system, not the object, but often we can think of it associated with the object only, and forget about the systemWork and Potential EnergyFor every conservative force a potential energy function can be foundEvaluating the difference of the function at any two points in an object’s path gives the negative of the work done by the force between those two pointsGravitational Potential EnergyGravitational Potential Energy is the energy associated with the relative position of an object in space near the Earth’s surface Objects interact with the earth through the gravitational force Actually the potential energy is for the earth-object systemWork and Gravitational Potential Energy PE = mgy Units of Potential Energy are the same as those of Work and Kinetic EnergyfigravityPEPEW−=Reference Levels for Gravitational Potential Energy A location where the gravitational potential energy is zero must be chosen for each problem The choice is arbitrary since the change in the potential energy is the important quantity Choose a convenient location for the zero reference height often the Earth’s surface may be some other point suggested by the problem Once the position is chosen, it must remain fixed for the entire problemConservation of Mechanical Energy Conservation in general To say a physical quantity is conserved is to say that the numerical value of the quantity remains constant throughout any physical process In Conservation of Energy, the total mechanical energy remains constant In any isolated system of objects interacting only through conservative forces, the total mechanical energy of the system remains constant.Conservation of Energy, cont.Total mechanical energy is the sum of the kinetic and potential energies in the system Other types of potential energy functions can be added to modify this equationffiifiPEKEPEKEEE+=+=Potential Energy Stored in a SpringInvolves the spring constant, kHooke’s Law gives the force F = - k x F is the restoring force F is in the opposite direction of x k depends on how the spring was formed, the material it is made from, thickness of the wire, etc.Potential Energy in a SpringElastic Potential Energy related to the work required to compress a spring from its equilibrium position to some final, arbitrary, position x2skx21PE =Conservation of Energy Including a SpringThe PE of the spring is added to both sides of the conservation of energy equationThe same problem-solving strategies apply fsgisg)PEPEKE()PEPEKE(++=++Nonconservative Forces with Energy Considerations When nonconservative forces are present, the total mechanical energy of the system is not constant The work done by all nonconservative forces acting on parts of a system equals the change in the mechanical energy of the systemncW Energy= ∆Notes About Conservation of EnergyWe can neither create nor destroy energy Another way of saying energy is conserved If the total energy of the system does not remain constant, the energy must have crossed the boundary by some mechanism Applies to areas other than physicsWork Done by Varying Forces The work done by a variable force acting on an object that undergoes a displacement is equal to the area under the graph of F versus xSpring Example Spring is slowly stretched from 0 to xmax W = ½kx²applied restoring = - = kxF Fr rSpring Example, cont. The work is also equal to the area under the curve In this case, the “curve” is a triangle A = ½ B h gives W = ½ k
View Full Document
Unlocking...