MECHANICAL (TOTAL) ENERGY

Transcription

1 DO NOW: 1/19 If you haven t already, please take the short google form survey posted on Edmodo Please turn in your Work done by friction Lab in the top tray

2 POTENTIAL ENERGY Stored energy An object that has the potential to move Ex: a rock on the edge of a cliff, or an arrow ready to fire on a bent bow Symbol: U Two types: gravitational potential (U g ) and elastic potential energy (U s )

3 MECHANICAL (TOTAL) ENERGY Mechanical energy the sum of potential (elastic and gravitational) and kinetic energy for an object; also measured in Joules ME or E = KE + U g + U s

4 GRAVITATIONAL POTENTIAL ENERGY The stored energy of an object due to its position relative to a gravitational source U = mgh m=mass g=9.8 m/s 2 h=height above zero *Potential energy is measured relative to a position we call zero height is measured above this point

5 EXAMPLE 1: A 2 KG TEXTBOOK IS HELD 0.5 M ABOVE A TABLE TOP. A. HOW MUCH POTENTIAL ENERGY DOES THE BOOK HAVE RELATIVE TO THE TABLE TOP? B. IF THE TABLE IS 0.7 M HIGH, HOW MUCH POTENTIAL ENERGY DOES THE BOOK HAVE RELATIVE TO EARTH S SURFACE? C. IF THE RADIUS OF EARTH IS 6.37 X 10 6 M, WHAT IS THE POTENTIAL ENERGY OF THE BOOK RELATIVE TO THE CENTER OF EARTH?

6 ELASTIC POTENTIAL ENERGY Potential energy due to a stretched or compressed spring (elastic object) U s = 1 2 kx2 where k is a constant that tells you how elastic something is called the spring constant or force constant. k is measured in N/m x is the distance the spring is stretched or compressed measured in meters.

7 EXAMPLE 2: A SPRING WITH A SPRING CONSTANT OF 3.5 N/M HAS A RELAXED LENGTH OF 0.2M. WHEN IT IS STRETCHED TO A LENGTH OF 0.3M, A. HOW MUCH FORCE DOES THE SPRING EXERT? B. HOW MUCH ELASTIC POTENTIAL ENERGY DOES IT HAVE?

8 LAW OF CONSERVATION OF ENERGY In an isolated system, the total energy of the system remains constant. - Work done by Gravitational and Elastic forces cause energy to transform from kinetic to potential or vice versa - Work done by other forces (applied or friction) cause energy to be transferred from one object to another

9 SOME IMPORTANT DEFINITIONS: Conservative force a force that converts potential energy into kinetic energy (or vice versa) of an object, but doesn t change the overall mechanical energy. - Gravitational force - Elastic force (from a spring or other elastic object) Non-conservative force a force that adds or takes away total mechanical energy from the object. (Mechanical energy is NOT conserved). - Applied force - Friction (converts mechanical energy into thermal energy, object will lose energy)

10 LAW OF CONSERVATION OF ENERGY IN EQUATION FORM W NC = ME ME F = W NC + ME i KE f + U g,f +U s,f =W NC +KE i +U g,i +U s,i 1 2 mv f 2 + mgh f kx f 2 = W NC mv i 2 + mgh i kx i 2

11 CONSERVATIVE FORCES U g is being converted into KE In Equation form: KE = U g where U g : KE = U s where U s : U s is being converted into KE In Equation form: U s i + KE i = U s f + KE f U gi + KE i = U gf + KE f

12 NON-CONSERVATIVE FORCES The applied force from the bat does work on the ball, increasing its kinetic and gravitational potential energy. In equation form: W bat = ME where ME:+ KE f + U gf = KE i + U gi + W bat The friction force from the road does negative work on the car, decreasing its kinetic energy. In equation form: W fric = ME KE f = KE i W fric where ME:-

13 EXAMPLE 3 SITUATIONS For each example below, state if the mechanical energy IS or IS NOT conserved. State which forms of energy are increasing or decreasing Write an equation that represents the initial and final forms of energies.

14 EXAMPLE 3 SITUATIONS Energy IS conserved Ug is decreasing, KE is increasing U gi + KEi = Ug f + KEf Energy IS NOT conserved KE is increasing W NC = KEf

15 Initial forms of Energy Outside Work (nonconservative only) LOL CHARTS Final forms of Energy KE U g U s KE U g U s Bar graph sketch for types of energy before the interaction Arrow drawn into the circle if positive work is done by applied force Arrow drawn out of the circle if negative work is done by applied force or friction Bar graph sketch for types of energy after the interaction ME i + W NC = ME f

16 Initial forms of Energy Outside Work (nonconservative only) LOL CHARTS Final forms of Energy KE U g U s KE U g U s Bar graph sketch for types of energy before the interaction Arrow drawn into the circle if positive work is done by applied force Arrow drawn out of the circle if negative work is done by applied force or friction Bar graph sketch for types of energy after the interaction ME i + W NC = ME f

17 EXAMPLE LOL CHART A toy car initially is released from rest at the top of a grassy hill. It experiences friction as it rolls to the bottom of the hill. Initial forms of Energy Outside Work (nonconservative only) Final forms of Energy KE U g U s KE U g U s U g,i W fric = KE f

18 EXAMPLE LOL CHART A toy car initially is released from rest at the top of a grassy hill. It experiences friction as it rolls to the bottom of the hill. Initial forms of Energy Outside Work (nonconservative only) Final forms of Energy KE U g U s KE U g U s U g,i W fric = KE f

19 EXAMPLE 4: THE HIGHEST HILL OF A ROLLER COASTER IS 25 M TALL, FOR WHICH FRICTION IS NEGLIGIBLE. IF THE ROLLER COASTER AND PASSENGERS HAS A MASS OF 15,000 KG AND STARTS FROM REST AT THE TOP OF THE HILL. a.how fast is it moving when it gets to the bottom of the hill? b.the next hill is only 15 m tall. What is the speed of the roller coaster at the top of this hill?

20 Initial forms of Energy Outside Work (nonconservative only) LOL CHART FOR A Final forms of Energy KE U g U s KE U g U s Bar graph sketch for types of energy before the interaction Arrow drawn into the circle if positive work is done by applied force Arrow drawn out of the circle if negative work is done by applied force or friction Bar graph sketch for types of energy after the interaction U g,i + 0 = KE f

21 Initial forms of Energy Outside Work (nonconservative only) LOL CHART FOR B Final forms of Energy KE U g U s KE U g U s U g,i + 0 = KE f + U g,f

22 EXAMPLE 4 SOLUTION A. Initially: All energy is gravitational potential (U gi =mgh i ) mgh i =1/2 mv f 2 At the bottom of the first hill: All energy is kinetic since h=0 (KE f =1/2 mv f2 ) No applied or friction forces, so energy is conserved: ME i =ME f v f = 2gh i = 2 9.8m s 2 25m = 22.1 m/s B. Initially, ME i =U g =mgh i At the top of the second hill: Energy is kinetic and grav. potential (ME f =1/2 mv f2 +mgh f ) mgh i =1/2 mv f2 +mgh f v f = 2gh i 2gh f = 2 9.8m s 2 (25m 15m) = 14 m/s

23 EXAMPLE 5: A 10KG MASS COMPRESSES A SPRING (K=80 N/M) A DISTANCE OF 0.8M FROM ITS EQUILIBRIUM POSITION. IT IS RELEASED FROM REST AND SLIDES UP A FRICTIONLESS CURVED HILL AS SHOWN BELOW. a. How fast is the mass traveling when it loses contact with the spring? b. How high does the mass travel up the hill before it stops moving?

24 Initial forms of Energy Outside Work (nonconservative only) LOL CHART FOR A Final forms of Energy KE U g U s KE U g U s U s,i + 0 = KE f

25 Initial forms of Energy Outside Work (nonconservative only) LOL CHART FOR B Final forms of Energy KE U g U s KE U g U s U s,i + 0 = U g,f

26 EXAMPLE 5 SOLUTION A. Initially: energy is elastic potential only (ME i =1/2kx i2 ) At the end: spring isn t stretched, so energy is kinetic only (ME f =1/2mv f2 ) No friction or applied forces, so ME is conserved (ME i =ME f ) 1 kx 2 i 2 = 1 mv 2 f 2 v 2 f = kx i m 2 v f = kx i 2 80N = m (0.8m)2 m 10kg = 2.26m/s B. At the end, energy is gravitational potential only (U g =mgh f ) 1 2 kx i 2 = mgh f h f = =.5 80N m (0.8m)2 mg 10kg 9.8m/s kx i 2 =.261m

27 EXAMPLE 6: A 10.0 kg crate is pulled up a rough incline with an initial speed of 1.5 m/s. The pulling force is N parallel to the incline, which makes an angle of 15.0 with the horizontal. Assuming the coefficient of kinetic friction is 0.40 and the crate is pulled a distance of 7.5 m, find the following: a. The work done by the Earth s gravity on the crate. b. The work done by the force of friction on the crate. c. The work done by the puller on the crate. d. The change in kinetic energy of the crate. e. The speed of the crate after it is pulled 7.5 m.

28 Initial forms of Energy Outside Work (nonconservative only) LOL CHART Final forms of Energy KE U g U s KE U g U s KE i + W app W fric = KE f + U g,f

29 EXAMPLE 6 SOLUTION 15 C. W app =Fdcosθ where θ=0 W app =F a d W app =100N*7.5m=750J D. Since there are non-conservative forces, the total mechanical energy will change. We need to write an equation with energy: ME f = W NC + ME i U g,i + KE f = W app + W f + KE i KE f KE i = W app + W f U g,i = 750J + 284J 190J Change in kinetic energy=276j A. W g =-mg h Find h: sin 15 = h 7.5m h=1.94m W g =-10kg*9.8m/s2*1.94m= -190J B. W f =F k dcosθ where θ= 180 W f =-F k d Find F k : F k =µ k F N on a ramp, F N =mgcosθ F k =0.4*10kg*9.8m/s2*cos 15 = 37.9N W f =-37.9N*7.5m=-284J