Work and Energy Definition of work Examples. Definition of Mechanical Energy. Conservation of Mechanical Energy, Pg 1

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1 Work and Energy Definition of work Examples Work and Energy Today s Agenda Definition of Mechanical Energy Conservation of Mechanical Energy Conservative forces Conservation of Mechanical Energy, Pg 1

2 Give it a try: What work does gravity do as my book is moved upwards? (a) W = mgy (b) W = 0 (c) W = -mgy final mg y initial mg Conservation of Mechanical Energy, Pg 2

3 Work done by gravity Lets compute the work done by the gravitational force. final initial mg mg y W= F Cos d W = mg Cos 180 h = -mgy Conservation of Mechanical Energy, Pg 3

4 Give it a try: A crane lowers a girder into place at constant speed. Consider the work W g done by gravity and the work W T done by the tension in the cable. Which h is true? A. W g > 0 and W T > 0 B. W g > 0 and W T < 0 C. W g < 0 and W T > 0 D. W g < 0 and W T < 0 E. W g = 0 and W T = 0 Conservation of Mechanical Energy, Pg 4

5 Conservative forces For conservative forces the work done does not depend on path taken, only the starting and finishing points matter. Ex. gravity For conservative force work on a closed path is zero Ex. Roller coaster This semester Conservative forces Gravity Springs Non-conservative forces Anything else! Conservative forces give objects a potential energy!!!! Conservation of Mechanical Energy, Pg 5

6 Potential energy Potential energy: Stored energy Energy depends on the position or configuration of an object. Potential energy due to gravity Potential energy of a spring. U = -W conservative Conservation of Mechanical Energy, Pg 6

7 Work done by gravity Lets compute the work done by the gravitational force. final initial mg mg y f y i By moving the block up a distance y, it gains potential energy of mgy! U = -W C U = -(-mg(y f -y i ))= mg( y) Conservation of Mechanical Energy, Pg 7

8 Gravitational potential energy Potential energy due to earth s gravitational field Energy of Position Work by gravity W = -mg( y) Work is energy given to object! For an object at height y and mass m. U g = mgy + U g0 Usually we call U 0 = 0 U g = mgy y Conservation of Mechanical Energy, Pg 8

9 The Work-Energy Theorem W tot = K W C + W NC = K W NC = K W C W NC = K + U Conservation of Mechanical Energy, Pg 9

10 Mechanical energy The total mechanical energy is defined as the sum of the potential energy and the kinetic energy for an object. E mech = K + U W NC = K + U = (K f -K i ) + (U f -U i ) W NC = (K f + U f f) - (K i + U i i) W NC = E f -E i Conservation of Mechanical Energy, Pg 10

11 Conservation of Mechanical Energy So if only conservative forces are doing work (ie W NC is zero), the total mechanical energy of a system is conserved. W NC = E f - E i = 0 E initial iti = E final E = K + U is constant!!! Both K and U can change, but E = K + U remains constant. Conservation of Mechanical Energy, Pg 11

12 Work by a Non-constant Force 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 x Physics 201: Lecture 13, Pg 12

13 Work by a Non-constant Force 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 x F F = -kx W = ½ (-kx 1 )(x 1 ) W =-½k(x 2 1 ) x 1 x W C = - U -kx 1 U = ½ kx 2 Physics 201: Lecture 13, Pg 13

14 Clicker Question 4: You grasp the end of a spring that is attached to the wall and is initially in its resting position. You pull it out until it is extended 0.1 m from its resting position, then push it in until it is compressed by 0.1 m from its resting position. Finally, you return the spring to its resting position. The spring constant is k = 20 N/m. The total work W done by the spring on your hand is (a) W < 0 (b) W = 0 (c) W > 0 Physics 131: Lecture 15, Pg 14

15 New Conservation of Mechanical Energy Good when only gravity or a spring-like force are doing work on an object E f = E i K f + U f =K i + U i ½k s f2 + ½mv f2 + mgh f = ½k s i2 + ½mv i2 + mgh i Physics 201: Lecture 13, Pg 15

16 DEFINITION OF AVERAGE POWER Average power is the rate at which work is done, and it is obtained by dividing the work by the time required to perform the work. P AVG E t Joule s Watt (W) Physics 201: Lecture 13, Pg 16

17 Give it a try: Four students run up the stairs in the time shown. Which student has the largest power output? Physics 201: Lecture 13, Pg 17