Short Version : 6. Work, Energy & Power

Usefulness for Studying Energy Alternative ways to describe motion: –Conservation of energy. –Hamiltonian formulism. Extension to non-mechanical studies: – Thermal physics. – Electrodynamics.

6.1. Work Work W done on an object by a constant force F is  r F = displacement along direction of F. Note: F need not be a net force.  F r = force along direction of  r.

Work & the Scalar Product Work is a scalar. Scalar = quantity specified by a single number that is the same in every coordinate system. Scalar has no direction. Scalar (dot) product of vectors A & B : is a scalar  = angle between F &  r A B B A = B cos   3-D 2-D

Proof  Letbe the unit vectors along the Cartesian axes.   C A + C A ABAB   CBCB ( A + C ) B

6.2. Forces that Vary

Tactics 6.1. Integrating   Example: Sincewe have See Appendix A for integral table.

Stretching a Spring

Example 6.4. Bungee Jumping Bungee cord is 20 m long with k = 11 N/m. At lowest point, cord length is doubled. (a) How much work is done on cord? (b) How does work done in the last meter compare with that done in the 1 st meter? (a) (b) 1 st meter Last meter

Force & Work in 2- & 3- D Line integral 1-D 2-D 3-D

Work Done Against Gravity Only vertical displacement requires work against gravity W = m g h

6.3. Kinetic Energy Kinetic energy : K is relative (depends on reference frame). K is a scalar. Work-energy theorem :

GOT IT? 6.3. For each situation, tell whether the net work done on a soccer ball is positive, negative, or zero. Justify your answer using the work-energy theorem. (a)You carry the ball to the field, walking at constant speed. (b) You kick the stationary ball, starting it flying through the air. (c) The ball rolls along the filed, gradually coming to a halt. zero (  K=0) positive (  K>0) negative (  K<0)

Energy Units [energy] = [work]= J (SI) CGS: Other energy units: eV (electron-volt): used in nuclear, atomic, molecular, solid state physics. cal (calorie), BTU (British Thermal Unit): used in thermodynamics. kW-h (kilowatt-hours): used in engineering. See Appendix C

6.4. Power Average power : (Instantaneous) power : Example 6.7. Climbing Mount Washington A 55 kg hiker makes the vertical rise of 1300 m in 2 h. A 1500 kg car takes ½ h to go there. Neglecting loss to friction, what is the average power output for each. Hiker: Car:

Energy and Society 2008: 4.7  J or 1.5  W

Power & Velocity Example 6.9. Bicycling Riding a 14 kg bicycle at a steady 18 km/h (5.0 m/s), you experience a 30 N force from air resistance. If you mass 68 kg, what power must you supply (a)on level ground. (b) up a 5  slope. (a) (b) F air FgFg v v