Introduction to Mechanics Energy Conservation

Transcription

1 Introduction to Mechanics Energy Conservation Lana Sheridan De Anza College Mar 22, 2018

2 Last time conservative forces and potential energy energy diagrams mechanical energy energy conservation

3 Overview energy conservation more practice with energy conservation

4 Energy Conservation Energy conservation for a system can be expressed as: W ext = K + U The work done by external forces includes work done by nonconservative forces and applied forces.

5 Isolated System: Energy Conservation A rock is dropped from rest a height h. By considering energy, find an expression for how fast it is moving just before it hits the ground.

6 Isolated System: Energy Conservation A rock is dropped from rest a height h. By considering energy, find an expression for how fast it is moving just before it hits the ground. System: rock + Earth Let point i be the moment it is dropped; f be just before it strikes the ground. Let y = 0 be the ground level. K + U = 0 0 (K f K i ) + ( U 0 f U i ) = 0 K f = U i 1 2 mv 2 = mgh v = 2gh

7 For the gravitational situation of the falling book, Equa 1 Isolated System: Energy Conservation 2mv f 2 1 mgy f 1 2mv i 2 1 mgy i As the book falls to the Earth, the book Earth system gains kinetic energy such that the total of the two type Quick Quiz Three identical balls are thrown from the top of a building, all with the same constant: initial speed. E total,i As E total,f shown,. the first is thrown horizontally, the second at some angle above the horizontal, and the third at some angle below the horizontal. Neglecting air resistance, rank the speeds of the balls at the energy of the system as instant each hits the ground, from largest to smallest. The total energy of an isolated system is conserved. 2 1 If there are nonconservative forces acting within the is transformed to internal energy as discussed in Sect forces act in an isolated system, the total energy of the sy the mechanical energy is not. In that case, we can e DE system 0 where E system includes all kinetic, potential, and interna the most general statement of the energy version of the equivalent to Equation 8.2 with all terms on the right-ha 3 (A) 2, 1, 3 (B) 3, 1, 2 Q uick Quiz 8.3 A rock of mass m is dropped to the gro second rock, with mass 2m, is dropped from the same rock strikes (C) 1, the 2, ground, 3 what is its kinetic energy? (a) (b) four times that of the first rock (c) the same as tha as much (D) as that all the of the same first rock (e) impossible to dete Q uick Quiz 8.4 Three identical balls are thrown from with the same initial speed. As shown in Figure 8.3, t Figure 8.3 (Quick Quiz 8.4) zontally, the second at some angle above the horizont 2 Three identical balls are thrown Adapted from Serway & Jewett, angle page below 216. the horizontal. Neglecting air resistance,

8 For the gravitational situation of the falling book, Equa 1 Isolated System: Energy Conservation 2mv f 2 1 mgy f 1 2mv i 2 1 mgy i As the book falls to the Earth, the book Earth system gains kinetic energy such that the total of the two type Quick Quiz Three identical balls are thrown from the top of a building, all with the same constant: initial speed. E total,i As E total,f shown,. the first is thrown horizontally, the second at some angle above the horizontal, and the third at some angle below the horizontal. Neglecting air resistance, rank the speeds of the balls at the energy of the system as instant each hits the ground, from largest to smallest. The total energy of an isolated system is conserved. 2 1 If there are nonconservative forces acting within the is transformed to internal energy as discussed in Sect forces act in an isolated system, the total energy of the sy the mechanical energy is not. In that case, we can e DE system 0 where E system includes all kinetic, potential, and interna the most general statement of the energy version of the equivalent to Equation 8.2 with all terms on the right-ha 3 (A) 2, 1, 3 (B) 3, 1, 2 Q uick Quiz 8.3 A rock of mass m is dropped to the gro second rock, with mass 2m, is dropped from the same rock strikes (C) 1, the 2, ground, 3 what is its kinetic energy? (a) (b) four times that of the first rock (c) the same as tha as much as that of the first rock (e) impossible to dete (D) all the same Q uick Quiz 8.4 Three identical balls are thrown from with the same initial speed. As shown in Figure 8.3, t Figure 8.3 (Quick Quiz 8.4) zontally, the second at some angle above the horizont 2 Three identical balls are thrown Adapted from Serway & Jewett, angle page below 216. the horizontal. Neglecting air resistance,

9 s E 0 = mgh. If friction and other nonconservative forces can be ignored, s mechanical energy remains fixed at throughout its motion. Thus, Energy Conservation and Energy Diagrams E 0 E = U + K = E 0 he ballimage moves, aits ball potential on a track, energyreleased falls andfrom risesrest in the at same pointway A. Suppose as the we ter all, can the gravitational ignore friction potential and air energy, resistance. U = mgy, is directly proportional simply con potential en coaster). In cally surfac ing much o dissipated energy, suc h y A D B C x FIGURE frictionless The ball sta zero speed At D, wher returns to z The ball speeds up as it drops lower.

10 Energy Conservation and Energy Diagrams ERGY AND CONSERVATION OF ENERGY otential rack This can be represented with a potential energy curve of the same shape! y curve of the hanical e, e height it follows urve up Note that anishes oints of E 0 = mgh Energy A B K K U C K U D U x The lower points correspond to less potential energy and more kinetictoenergy. the height of the track, y.inasense, then, the track itself represents a grap corresponding potential energy. If the initial This PEis is shown E 0, then explicitly K + in U Figure = E 0 at 8 11, all points. where we plot energy on the axis and x on the horizontal axis. The potential energy U looks just like the

11 Energy Conservation Diagrams Examples Section 8 Potential Energy Curves and Equipotentials An object 41. moves Figure along 8 24 the shows x axis, a potential subject energy to thecurve potential as a function energy of shown. The x. object In qualitative has a mass terms, of describe 1.1 kg and the subsequent starts at rest motion at point of an object that starts at rest at point A. A. Which points are stable equilibria? 10.0 J U A E 6.0 J.0 J C D 2.0 J (A) A and E x 1.0 m 2.0 m 3.0 m 4.0 m.0 m (B) B and D FIGURE 8 24 Problems 41, 42, 43, and 46 (C) C (D) B, C, 42. and An D object moves along the x axis, subject to the potential en- B

12 Energy Conservation Diagrams Examples Section 8 Potential Energy Curves and Equipotentials An object 41. moves Figure along 8 24 the shows x axis, a potential subject energy to thecurve potential as a function energy of shown. The x. object In qualitative has a mass terms, of describe 1.1 kg and the subsequent starts at rest motion at point of an object that starts at rest at point A. A. Which points are stable equilibria? 10.0 J U A E 6.0 J.0 J C D 2.0 J B (A) A and E x 1.0 m 2.0 m 3.0 m 4.0 m.0 m (B) B and D FIGURE 8 24 Problems 41, 42, 43, and 46 (C) C (D) B, C, 42. and An D object moves along the x axis, subject to the potential en-

13 Energy Conservation Diagrams Examples Section 8 Potential Energy Curves and Equipotentials An object 41. moves Figure along 8 24 the shows x axis, a potential subject energy to thecurve potential as a function energy of shown. The x. object In qualitative has a mass terms, of describe 1.1 kg and the subsequent starts at rest motion at point of an object that starts at rest at point A. A. Which points are unstable equilibria? 10.0 J U A E 6.0 J.0 J C D 2.0 J (A) A and E x 1.0 m 2.0 m 3.0 m 4.0 m.0 m (B) B and D FIGURE 8 24 Problems 41, 42, 43, and 46 (C) C (D) B, C, 42. and An D object moves along the x axis, subject to the potential en- B

14 Energy Conservation Diagrams Examples Section 8 Potential Energy Curves and Equipotentials An object 41. moves Figure along 8 24 the shows x axis, a potential subject energy to thecurve potential as a function energy of shown. The x. object In qualitative has a mass terms, of describe 1.1 kg and the subsequent starts at rest motion at point of an object that starts at rest at point A. A. Which points are unstable equilibria? 10.0 J U A E 6.0 J.0 J C D 2.0 J B (A) A and E 1.0 m 2.0 m 3.0 m 4.0 m.0 m x (B) B and D (C) C FIGURE 8 24 Problems 41, 42, 43, and 46 (D) B, C, 42. and An D object moves along the x axis, subject to the potential en-

15 Energy Conservation Diagrams Examples Section 8 Potential Energy Curves and Equipotentials An object 41. moves Figure along 8 24 the shows x axis, a potential subject energy to thecurve potential as a function energy of shown. The x. object In qualitative has a mass terms, of describe 1.1 kg and the subsequent starts at rest motion at point of an object that starts at rest at point A. A. Which points are turning points? 10.0 J U A E 6.0 J.0 J C D 2.0 J (A) A and E x 1.0 m 2.0 m 3.0 m 4.0 m.0 m (B) B and D FIGURE 8 24 Problems 41, 42, 43, and 46 (C) C (D) B, C, 42. and An D object moves along the x axis, subject to the potential en- B

16 Energy Conservation Diagrams Examples Section 8 Potential Energy Curves and Equipotentials An object 41. moves Figure along 8 24 the shows x axis, a potential subject energy to thecurve potential as a function energy of shown. The x. object In qualitative has a mass terms, of describe 1.1 kg and the subsequent starts at rest motion at point of an object that starts at rest at point A. A. Which points are turning points? 10.0 J U A E 6.0 J.0 J C D 2.0 J B (A) A and E 1.0 m 2.0 m 3.0 m 4.0 m.0 m x (B) B and D (C) C FIGURE 8 24 Problems 41, 42, 43, and 46 (D) B, C, 42. and An D object moves along the x axis, subject to the potential en-

17 m k = 0.60, what is the force constant of the spring? Energy Conservation Diagrams Examples An Section object 8 moves Potential along the Energy x axis, Curves subject and to Equipotentials the potential energy 41. Figure 8 24 shows a potential energy curve as a function of shown. The object has a mass of 1.1 kg and starts at rest at x. In qualitative terms, describe the subsequent motion of an point object A. that starts at rest at point A. (a) What is the object s speed at point B? (b) At point C? 10.0 J 6.0 J.0 J U A C D E General 0. IP hill. I the b a spe hill, i 9.00 m of the 1. In 2. A has a the sk the n Ignor 2.0 J B 1.0 m 2.0 m 3.0 m 4.0 m.0 m FIGURE 8 24 Problems 41, 42, 43, and 46 x

18 m k = 0.60, what is the force constant of the spring? Energy Conservation Diagrams Examples An Section object 8 moves Potential along the Energy x axis, Curves subject and to Equipotentials the potential energy 41. Figure 8 24 shows a potential energy curve as a function of shown. The object has a mass of 1.1 kg and starts at rest at x. In qualitative terms, describe the subsequent motion of an point object A. that starts at rest at point A. (a) What is the object s speed at point B? (b) At point C? 10.0 J 6.0 J.0 J U A C D E General 0. IP hill. I the b a spe hill, i 9.00 m of the 1. In 2. A has a the sk the n Ignor 2.0 J 1.0 m B 2.0 m 3.0 m 4.0 m FIGURE 8 24 Problems 41, 42, 43, and 46 Hypoth: greater at B than C.0 m x

19 m k = 0.60, what is the force constant of the spring? Energy Conservation Diagrams Examples An Section object 8 moves Potential along the Energy x axis, Curves subject and to Equipotentials the potential energy 41. Figure 8 24 shows a potential energy curve as a function of shown. The object has a mass of 1.1 kg and starts at rest at x. In qualitative terms, describe the subsequent motion of an point object A. that starts at rest at point A. (a) What is the object s speed at point B? (b) At point C? 10.0 J 6.0 J.0 J 2.0 J U A 1.0 m B 2.0 m C 3.0 m D 4.0 m E.0 m FIGURE 8 24 Problems 41, 42, 43, and 46 Hypoth: greater at B than C (a) v = 3.8 m/s ; (b) v = 2.7 m/s x General 0. IP hill. I the b a spe hill, i 9.00 m of the 1. In 2. A has a the sk the n Ignor

20 How to Solve Energy Conservation Problems 1 Draw (a) diagram(s). Free body diagrams or full pictures, as needed. 2 Make a hypothesis or estimate of what the answer will be. 3 Identify the system. State what it is. Is it isolated? 4 Identify the initial point / configuration of the system. Identify the final point / configuration of the system. 6 Write the energy conservation equation. 7 Fill in the expressions as needed. 8 Solve. 9 Analyze answer: reasonable value?, check units, etc.

21 Energy Conservation Example als, the oils you just way hile nds of me oor. for calthe tion the Section 8.2 Analysis Model: Isolated System (Energy) 3. A block of mass 0.20 kg is placed on top of a light, vertical spring of force constant 000 N/m and pushed W downward so that the spring is compressed by m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise? 4. A 20.0-kg cannonball is fired from a cannon with muzzle speed of m/s at an angle of with the hor- W izontal. A second ball is fired at an angle of Use the isolated system model to find (a) the maximum height reached by each ball and (b) the total mechanical energy of the ball Earth system at the maximum height for each ball. Let y 0 at the cannon. h. Review. A bead slides without friction around a loop-the-loop (Fig. P8.). The bead is released from AMT M R

22 Energy Conservation Example als, the oils you just way hile nds of me oor. for calthe tion the Section 8.2 Analysis Model: Isolated System (Energy) 3. A block of mass 0.20 kg is placed on top of a light, vertical spring of force constant 000 N/m and pushed W downward so that the spring is compressed by m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise? 4. A 20.0-kg cannonball is fired from a cannon with muzzle speed of m/s at an angle of with the hor- W W izontal. app U A s = 1 second 2 kx 2 K.E. & Grav P.E. U ball is fired at an angle of g = mgh Use the isolated system model to find (a) the maximum height reached by each ball and (b) the total mechanical energy of the ball Earth system at the maximum height for each ball. Let y 0 at the cannon. h. Review. A bead slides without friction around a loop-the-loop (Fig. P8.). The bead is released from AMT M R

23 Energy Conservation Example als, the oils you just way hile nds of me oor. for calthe tion the Section 8.2 Analysis Model: Isolated System (Energy) 3. A block of mass 0.20 kg is placed on top of a light, vertical spring of force constant 000 N/m and pushed W downward so that the spring is compressed by m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise? 4. A 20.0-kg cannonball is fired from a cannon with muzzle speed of m/s at an angle of with the hor- W W izontal. app U A s = 1 second 2 kx 2 K.E. & Grav P.E. U ball is fired at an angle of g = mgh Use the isolated system model to find (a) the maximum System: height block reached + spring by + each Earth. ball and (b) the total mechanical energy of the ball Earth system at the maximum height for each ball. Let y 0 at the cannon.. Review. A bead slides without friction around a loop-the-loop (Fig. AMT h R M P8.). The bead is released from

24 Energy Conservation Example als, the oils you just way hile nds of me oor. for calthe tion the Section 8.2 Analysis Model: Isolated System (Energy) 3. A block of mass 0.20 kg is placed on top of a light, vertical spring of force constant 000 N/m and pushed W downward so that the spring is compressed by m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise? 4. A 20.0-kg cannonball is fired from a cannon with muzzle speed of m/s at an angle of with the hor- W W izontal. app U A s = 1 second 2 kx 2 K.E. & Grav P.E. U ball is fired at an angle of g = mgh Use the isolated system model to find (a) the maximum System: height block reached + spring by + each Earth. ball and (b) the total mechanical energy of the ball Earth system point, at i : the release maximum point (max height compression for of spring), Initial choose each ball. y = 0, Let U y = 0 at this the cannon. point Final. point, Review. f : A point bead of slides max without height offric- tion around a loop-the-loop (Fig. block AMT System M P8.). is isolated. The bead is released from h R

25 Energy Conservation Example als, the oils you just way hile nds of me oor. for calthe tion the Section 8.2 Analysis Model: Isolated System (Energy) 3. A block of mass 0.20 kg is placed on top of a light, vertical spring of force constant 000 N/m and pushed W downward so that the spring is compressed by m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise? 4. A 20.0-kg cannonball is fired from a cannon with muzzle W speed of m/s at K an angle + U of = with 0 the hor- izontal. A second ball is fired at an angle of Use the isolated system model to find (a) the maximum height reached by each ball and (b) the total mechanical energy of the ball Earth system at the maximum height for each ball. Let y 0 at the cannon.. Review. A bead slides without friction around a loop-the-loop (Fig. AMT h R M P8.). The bead is released from 1 Problem rest at from a height Serway h & Jewett, 3.0R. 9th(a) ed, What page 236. Figure P8.

26 Energy Conservation Example als, the oils you just way hile nds of me oor. for calthe tion the Section 8.2 Analysis Model: Isolated System (Energy) 3. A block of mass 0.20 kg is placed on top of a light, vertical spring of force constant 000 N/m and pushed W downward so that the spring is compressed by m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise? 4. A 20.0-kg cannonball is fired from a cannon with muzzle speed of m/s at K an angle + U of = with 0 the hor- W izontal. A second ball is fired at an angle of Use the ( 0 0 K f isolated K i ) + ( U system 0 s,f Umodel s,i ) + (Uto g,f find U (a) 0 g,i ) the = maximum 0 height reached by each ball and (b) Uthe g,f total = Umechani- cal energy of the ball Earth sys- s,i tem at the maximum height for mgh = 1 2 kx 2 each ball. Let y 0 at the cannon.. Review. A bead slides without friction around a loop-the-loop (Fig. h = 10.2 m h = kx 2 2mg R AMT M P8.). The bead is released from 1 Problem rest at from a height Serway h & Jewett, 3.0R. 9th(a) ed, What page 236. Figure P8.

27 Energy Conservation: Example 8-10 ENERGY A block of mass m 1 = 2.40 kg is connected to a second block of mass m 2 = 1.80 kg. When the blocks are released from rest, they move through a distance d = 0.00 m, at which point m 2 hits the floor. mgiven 2 = 1.80 that kg, as the shown coefficient here. When of kinetic the blocks friction are released between m 1 and the horizontal m 2 hits the surface floor. Given is µ k that = the 0.40, coefficient find the of kinetic speedfric- of the blocks just before m 2 lands. m 2 lands. lock of mass t which point 0, find the speed of the blocks just before ravitapotens; it is e of h. h i and m 1 i d f y h in this lculate e nonrms of There- i f m 2 d 0

28 hud Example 8-10 ted to a second block of mass m 2 = 1.80 kg, as shown here. When the blocks are released e d = 0.00 m, at which point m 2 hits the floor. Given that the coefficient of kinetic fricface is m k = 0.40, find the speed of the blocks just before m 2 lands. therefore, the gravitaen it lands. The potenring this process; it is o know the value of h. nding points with i and m 1 i d f y h ) is doing work in this hus, we must calculate d E f, but also the nonn be written in terms of before m 2 lands. Therelve for the final speed. clude contributions hat E f depends, W nc. Recall that the g, and that it points E i = U i + K i = m 1 gh + m 2 gd W nc = -f k d = -m k m 1 gd m 2 i d f 0 System: Masses m 1 and m 2, modeled as point particles, and the Earth. U i = m 1 gh K i = 1 2 m # + m 2 gd W 1 2 m # 2 0 ext 2 = 0= K + U Here: U f = m 1 gh + 0 W ext = W nc = f k d K f = 1 2 m 1v m 2v K is the change 2 E f = U f in K.E. + K f = m 1 of both masses gh m 1v m 2v 2 U is the change in Grav. P.E. of the masses (only m 2 s changes)

29 Example 8-10 Points i and f are as labelled in the diagram. W ext = K + U W ext = (K 1,f + K 2,f 0 K 1,i 0 K 2,i ) + ( 0 U f U i ) f k d = ( 1 2 (m 1 + m 2 )v 2 0) + (0 m 2 gd) v = 2(m 2 µ k m 1 )gd m 1 + m 2 = 1.30 m/s

30 Summary energy conservation practice Final Exam Thursday, Mar 29, 9:1-11:1am, S3 (here). Homework Walker Physics: PREV: Ch 8, onward from page 243. Questions: 11, 13; Problems: 21, 23, 37, 41, 47,, 7, 9, 87, 9 NEW: Ch 8, Problem: 97 (Only 1 new problem. Notice in this problem, the ramp is curved, so the acceleration of the block is not constant, which means you cannot use kinematics equations. You must use conservation of energy.)