Phy 211 General Physics I Chapter 7 Kinetic Energy Work Lecture Notes What is Energy Energy is a scalar quantity associated with the state of an object or system of objects Energy is a calculated value that appears in nature and whose total quantity in a system always remains constant and accounted for Energy is said to be Conserved Loosely speaking energy represents the fuel necessary for changes to occur in the universe and is often referred to as the capacity to perform work We can think of energy as the currency associated with the transactions forces that occur in nature In mechanical systems energy is spent as force transactions are conducted Alternatively the exertion of force requires an expenditure of energy The SI units for energy are called Joules J In honor of James Prescott Joule 1 James Prescott Joule 1818 1889 English inventor scientist Interested in the efficiency of electric motors Described the heat dissipated across a resistor in electrical circuits now known as Joules Law Demonstrated that heat is produced by the motion of atoms and or molecules Credited with establishing the mechanical energy equivalent of heat Participated in establishing the Law of Energy Conservation Review The Scalar Dot Product Two vectors A and B can be multiplied to product a scalar resultant called the scalar or Dot product When using the magnitudes of the vectors A B A B cos where is the angle between vectors A and B When using vector components A B A x Bx A y By Useful properties of scalar products 1 Commutative property A A B B 2 Squaring vectors A A A 2 1 3 Unit vectors i i j j k k Important Application We will use the scalar product of the vectors of force and displacement to calculate work performed by force 2 Kinetic Energy The energy associated with an object s state of motion Kinetic energy is a scalar quantity that is never negative in value Definition 2 1 m Scalar form K mv2 SI units kg 2 s 1 1 1 1 Vector form K mv v mv2x mv2y mv2z 2 2 2 2 K K x Ky Kz Key Notes 1 The kinetic energy for the x y and z components are additive 2 Kinetic energy is relative to the motion of observer s reference frame since speed and velocity are as well 3 An object s kinetic energy depends more on its speed than its mass 4 Any change in an object s speed will affect a change in its kinetic energy 2 m 5 Unit comparison 1 J 1 kg s Work The energy transferred to from an object by the exertion of a force Work is essentially a measure of useful physical output or Work Effort x Outcome r F Definition standard form W or W F r Fx x Fy y Fz z F r cos SI Units for Work N m F r Notes 1 Unit comparison 1 J 1 N m 3 Work cont The general definition of Work for a varying as well as W r constant force W 0 dW ro F dr Graphically work is the area beneath the Force vs Displacement graph Fcos Fcos Area Area r r dW F dr W W r dW F dr 0 ro Work Kinetic Energy The net work performed on an object is related to the net force WNet FNet r m a r When FNet 0 a change in state of motion kinetic energy is implied WNet FNet dr K Ko K Derivation WNet dW WNet m a dr WNet WNet FNet dr dr m dv dt 1 m v v 2 m a dr dv dv dr m dt dr m dr dt dr 1 m v dv m v v vvo 2 1 m v o vo The Work Energy Theorem 2 4 Work Performed by Gravitational Force For a falling body no air drag Fg mg Wy Fg y mg y since cos 1 Wx Fg x 0 since cos 0 Wg Wx Wy mg y y Gravitational force only performs work in the vertical direction 1 W g is when y is 2 W g is when y is Fg mg What about gravitational force on an incline y Work Performed during Lifting Lowering Consider Joey blasting his pecs with a bench press workout assume vlift constant mg Given mbar 100 kg mbar g 980 N j y 1m j y FLift Applying Newton s 2nd Law FNet FLift mbar g j mbar a 0 FLift mbar g 980 N The Work performed Wg mbar g y 980 N m WLift FLift y 980 N m WNet FNet y WLift Wg 0 5 Work Performed by Elastic Force The elastic or restoring force associated with the compression or stretching of a simple spring increases linearly with the amount of deformation this is Hooke s Law The following assumptions are made The spring is ideal massless The spring obeys Hooke s Law Fspring k rstretch kx where rstretch x x where xo 0 is the equilibrium position of the spring and k is the spring constant in N m Work Performed by Elastic Force cont Consider a spring k 10 N m attached to a wall at one end and a mass 1 kg at the other The block is free to slide on a frictionless surface The block spring system is compressed 0 5 m then released m Fspring m m The work performed by the spring force is Wspring Fspring dx Fspring Equilibrium Position x 1 1 Wspring kx dx k x x xxo k x2 x20 xo 2 2 1 1 Wspring kx20 kx2 2 2 6 Power A measure of work effectiveness The time rate of energy transfer work due to an exerted force dW P dt W F r P Average Power avg t t dW F d r Instantaneous Power P dt dt WNet FNet dr K Average Net Power P Net t t t F dr dt SI units The Watt 1 W 1 J s Note Power is also related to Force Velocity dW F dr P F v dt dt Power cont The same work output can be performed at various power rates Example 1 Consider 100 J of work output accomplished over 2 different time intervals 100 J over 1 s 100 J 100 J over 100 s 100 J 100 W P P 1 W 1s 100 s Example 2a An 900 kg automobile accelerates from 0 to 30 m s in 5 8 s What is the average net power K K o Pnet t 1 2 900 kg 30 ms 0 ms 2 5 8 s 2 6 98 104 W or 93 6 hp Example 2b At the 30 m s how much force does the road exert on this vehicle Use the same power as 2a P 6 98 104 W P F v 6 98 104 W F 2330 N v 30 ms 7