Introduction to Mechanics Work and Energy

Transcription

1 Introuction to Mechanics Work an Energy Lana Sherian De Anza College Mar 15, 2018

2 Last time non-uniform circular motion an tangential acceleration energy an work

3 Overview energy work a more general efinition of work net work work one by a varying force(?)

4 Energy Energy is a ifficult concept to efine, but it is very important for physics. Energy can take many ifferent forms. Knowing the amount of energy a system has can tell us about what states or configurations we can fin the system in.

5 Energy Energy is a ifficult concept to efine, but it is very important for physics. Energy can take many ifferent forms. Knowing the amount of energy a system has can tell us about what states or configurations we can fin the system in. One way that energy is often escribe is that it represents the ability of a system to o work. We nee to know what work is!

6 Work Work is an amount of energy. The amount of work, W, one on an object epens on the applie force an the isplacement of the object as the force acts.

7 Work we o work. The greater the force, the greater the work; the grea tance, the greater the work. These simple ieas form the basis for ou of work.! Work is an To amount be specific, of energy. suppose we push a box with a constant force a Figure 7 1. If we move the box in the irection of F! F, through a isplace work W we have one is F: The amount of work, W, one on an object epens on the appliedefinition force anof the Work, isplacement W, When a Constant of the object Force as Is in the the force Direction acts. of Displac W = F If the force is in the same irection as the isplacement, SI unit: newton-meter 1N # m2 = joule, J W = F rce in hrough ase, t are in on the F F

8 Work W = F Units of Work? They have a special name: Joules, symbol J. 1 joule = 1 J = 1 Nm Work is not a vector. Work is a scalar.

9 The work one by the agent on the gravitational force book Earth system is rest through a ver What is the work one in mgy lifting f mgy a 3.0-kg i. book 0.50 m? to our iscussion o appear as an incre Physics the work an is at the kinetic energy S F Because the en S app r the work-kinetic e y Physics f appear as some fo book, we coul rel S mg y i (an therefore, the that was one in l tem ha the potenti allowe to fall. Th Figure 7.15 is release potenti An external agent lifts a book slowly from a height y i only be associate to a height y f. The amount of po the system. Movin may change the co Example

10 The work one by the agent on the gravitational force book Earth system is rest through a ver What is the work one in mgy lifting f mgy a 3.0-kg i. book 0.50 m? to our iscussion o appear as an incre Physics the work an is at the kinetic energy S F Because the en S app r the work-kinetic e y Physics f appear as some fo book, we coul rel S mg y i (an therefore, the that was one in l tem ha the potenti allowe to fall. Th Figure 7.15 is release potenti An external agent To lift a book withlifts constant a book slowly velocity, from requires a height Fy i app = only mg be associate to a height y f. The amount of po W = F = (3.0 kg)(9.81 m/s 2 )(.50 m) = 14.7 the Jsystem. Movin may change the co Example

11 Work To be specific, suppose we push a box with a constant force a Figure 7 1. If we move the box in the irection of F! F, through a isplace work W we have one is F: Definition of Work, W, When a Constant Force Is in the Direction of Displac If the force, W = F F, is in the same irection as the isplacement, the work one by F is given by SI unit: newton-meter 1N # m2 = joule, J W = F rce in hrough ase, t are in on the F F Note that work is the prouct of two magnitues, an hence it is a sca tion, notice that a small force acting over a large istance gives the sa

12 gest that when Work, we must cone of the force. nt represents a N through the rtant. Moving ving it 2 cm if What if the force is not in the irection of the isplacement? system) unert force of magt. one by the hammer on the nail ientifies the nail as the system, an the force from the hammer represents the influence from the environment. r orce on the itue Dr of Figure 7.2 An object unergoes We nee to exten our efinition of work. u, where u is u S F S a isplacement D r S uner the action of a constant force F S. (7.1) Work one by a

13 Work F is zero; W = F cos 90 = 0. This result leas naturally to an alternative way to think about the expression W = F cos u. In Figure 7 3 we show the isplacement an the force for the suitcase gle p at. The of e by F F cos F F cos For a constant applie force, Work is efine as: W = F = F cos θ

14 Work In this expression: W = F = F cos θ we use something calle the ot prouct of two vectors A an B: A B = AB cos θ

15 190 6 u Units of Work Thus, whenever we calculate work, we must be careful about its sign an assume it to be positive. Work can be positive or negative! the e- nerro gles F F F 90 < < 90 = ± < < 270 W = F cos θ > 0 (a) (b) (c) W = F cos θ = 0 W = F cos θ < 0 positive work zero work negative work For work one on a system: Positive energy is transferre to the system. Negative energy is transferre from the system.

16 Work one by 180 iniviual forces Chapter 7 Energy of a System S F is the only force that oes work on the block in this situation. S n mg S u r S S F have one a co tion, however, the chair, but oes not move the situation Also notice is zero when th application. Th Figure 7.3, the the gravitation icular to the tion of D S r. The sign of one by the ap is in the same If there are several Figure forces7.3 acting An on object a system, is is- each one can have an associate work. place on a frictionless, horizon-

17 Work one by iniviual forces 180 Chapter 7 Energy of a System S F is the only force that oes work on the block in this situation. S n mg S u have one a consierable tion, however, you have o the chair, but you o not oes not move through a the situation epicte in F Also notice from Equa is zero when the force ap application. That is, if u 5 Figure 7.3, the work one the gravitational force on icular to the isplaceme tion of D S r. The sign of the work al one by the applie force is in the same irection a the work one by the app Figure 7.3 An object is isplace on a frictionless, horizon- surface. W mg The = normal 0 Wforce F = S n F cos θ W n = 0tal an the gravitational force mg S o of that force is upwar, i In other wors, we can no work ask on what the object. is the work one on the system by application. When the pro each force separately. placement, W is negative. Pitfall Prevention 7.3 gravitational force on the r S S F

18 Net Work The net work is the sum of all the iniviual works. W net = i W i where W i = F i cos θ is the work one by the force F i. If the system can be moele as a particle (the only case we consier in this course): W net = F net cos θ assuming the net force is constant.

19 Question the car as it travels a istance along the roa. Picture the Problem Because f is the angle the slope makes with the horizontal, it is also the angle between was shown in Figure It follows that the angle between an the isplacement is N!! mg!!! mg! u u = 180. A car coasts the angle own between a hill that an makes is u = 90, anan angle the angle φ between to the F horizontal.! air an is F air N N = 90 mg Strategy For each force we calculate the work using where is the angle between ment! W = F cos u, u. The total work is the sum of the work one by each of the three forces. The work one by the weight (mg force) is (A) positive Solution (B) negative (C) zero 1. We start with the work one by the normal force, N!. From the figure we see that u = 90 for this force: 2. For the force of air resistance, u = 180 : (D) cannot be etermine W N = N cos u W air = F air cos 3. For gravity the angle u is u = 90 - f, as inicate in the W mg = mg cos figure. Recall that cos190 - f2 = sin f (see Appenix A):

20 Question the car as it travels a istance along the roa. Picture the Problem Because f is the angle the slope makes with the horizontal, it is also the angle between was shown in Figure It follows that the angle between an the isplacement is N!! mg!!! mg! u u = 180. A car coasts the angle own between a hill that an makes is u = 90, anan angle the angle φ between to the F horizontal.! air an is F air N N = 90 mg Strategy For each force we calculate the work using where is the angle between ment! W = F cos u, u. The total work is the sum of the work one by each of the three forces. The work one by the weight (mg force) is (A) positive Solution (B) negative (C) zero 1. We start with the work one by the normal force, N!. From the figure we see that u = 90 for this force: 2. For the force of air resistance, u = 180 : (D) cannot be etermine W N = N cos u W air = F air cos 3. For gravity the angle u is u = 90 - f, as inicate in the W mg = mg cos figure. Recall that cos190 - f2 = sin f (see Appenix A):

21 Question the car as it travels a istance along the roa. Picture the Problem Because f is the angle the slope makes with the horizontal, it is also the angle between was shown in Figure It follows that the angle between an the isplacement is N!! mg!!! mg! u u = 180. A car coasts the angle own between a hill that an makes is u = 90, anan angle the angle φ between to the F horizontal.! air an is F air N N = 90 mg Strategy For each force we calculate the work using where is the angle between ment! W = F cos u, u. The total work is the sum of the work one by each of the three forces. The work one by the normal force, N, is (A) positive Solution (B) negative (C) zero 1. We start with the work one by the normal force, N!. From the figure we see that u = 90 for this force: 2. For the force of air resistance, u = 180 : (D) cannot be etermine W N = N cos u W air = F air cos 3. For gravity the angle u is u = 90 - f, as inicate in the W mg = mg cos figure. Recall that cos190 - f2 = sin f (see Appenix A):

22 Question the car as it travels a istance along the roa. Picture the Problem Because f is the angle the slope makes with the horizontal, it is also the angle between was shown in Figure It follows that the angle between an the isplacement is N!! mg!!! mg! u u = 180. A car coasts the angle own between a hill that an makes is u = 90, anan angle the angle φ between to the F horizontal.! air an is F air N N = 90 mg Strategy For each force we calculate the work using where is the angle between ment! W = F cos u, u. The total work is the sum of the work one by each of the three forces. The work one by the normal force, N, is (A) positive Solution (B) negative (C) zero 1. We start with the work one by the normal force, N!. From the figure we see that u = 90 for this force: 2. For the force of air resistance, u = 180 : (D) cannot be etermine W N = N cos u W air = F air cos 3. For gravity the angle u is u = 90 - f, as inicate in the W mg = mg cos figure. Recall that cos190 - f2 = sin f (see Appenix A):

23 Question the car as it travels a istance along the roa. Picture the Problem Because f is the angle the slope makes with the horizontal, it is also the angle between was shown in Figure It follows that the angle between an the isplacement is N!! mg!!! mg! u u = 180. A car coasts the angle own between a hill that an makes is u = 90, anan angle the angle φ between to the F horizontal.! air an is F air N N = 90 mg Strategy For each force we calculate the work using where is the angle between ment! W = F cos u, u. The total work is the sum of the work one by each of the three forces. The work one by the air resistance (F air force) is (A) positive Solution (B) negative (C) zero 1. We start with the work one by the normal force, N!. From the figure we see that u = 90 for this force: 2. For the force of air resistance, u = 180 : (D) cannot be etermine W N = N cos u W air = F air cos 3. For gravity the angle u is u = 90 - f, as inicate in the W mg = mg cos figure. Recall that cos190 - f2 = sin f (see Appenix A):

24 Question the car as it travels a istance along the roa. Picture the Problem Because f is the angle the slope makes with the horizontal, it is also the angle between was shown in Figure It follows that the angle between an the isplacement is N!! mg!!! mg! u u = 180. A car coasts the angle own between a hill that an makes is u = 90, anan angle the angle φ between to the F horizontal.! air an is F air N N = 90 mg Strategy For each force we calculate the work using where is the angle between ment! W = F cos u, u. The total work is the sum of the work one by each of the three forces. The work one by the air resistance (F air force) is (A) positive Solution (B) negative (C) zero 1. We start with the work one by the normal force, N!. From the figure we see that u = 90 for this force: 2. For the force of air resistance, u = 180 : (D) cannot be etermine W N = N cos u W air = F air cos 3. For gravity the angle u is u = 90 - f, as inicate in the W mg = mg cos figure. Recall that cos190 - f2 = sin f (see Appenix A):

25 Question the car as it travels a istance along the roa. Picture the Problem Because f is the angle the slope makes with the horizontal, it is also the angle between was shown in Figure It follows that the angle between an the isplacement is N!! mg!!! mg! u u = 180. A car coasts the angle own between a hill that an makes is u = 90, anan angle the angle φ between to the F horizontal.! air an is F air N N = 90 mg Strategy For each force we calculate the work using where is the angle between ment! W = F cos u, u. The total work is the sum of the work one by each of the three forces. The net (or total) work one by all forces on the car is (A) positive Solution (B) negative (C) zero 1. We start with the work one by the normal force, N!. From the figure we see that u = 90 for this force: 2. For the force of air resistance, u = 180 : (D) cannot be etermine W N = N cos u W air = F air cos 3. For gravity the angle u is u = 90 - f, as inicate in the W mg = mg cos figure. Recall that cos190 - f2 = sin f (see Appenix A):

26 Question the car as it travels a istance along the roa. Picture the Problem Because f is the angle the slope makes with the horizontal, it is also the angle between was shown in Figure It follows that the angle between an the isplacement is N!! mg!!! mg! u u = 180. A car coasts the angle own between a hill that an makes is u = 90, anan angle the angle φ between to the F horizontal.! air an is F air N N = 90 mg Strategy For each force we calculate the work using where is the angle between ment! W = F cos u, u. The total work is the sum of the work one by each of the three forces. The net (or total) work one by all forces on the car is (A) positive Solution (B) negative (C) zero 1. We start with the work one by the normal force, N!. From the figure we see that u = 90 for this force: 2. For the force of air resistance, u = 180 : (D) cannot be etermine W N = N cos u W air = F air cos 3. For gravity the angle u is u = 90 - f, as inicate in the W mg = mg cos figure. Recall that cos190 - f2 = sin f (see Appenix A):

27 Work Done by a Variable Force Reasoning an We can unerstan that the work one by a force is the area A uner common mi the force-isplacement curve. the work requ rather than on Plotting a constant force F as a function of x ( x = ), F (x): To see how th work-energy t Force F Area = F = W work neee t fore, the work Answer: (c) W 2 = 3W 1 O x 1 x 2 Position FIGURE 7 6 Graphical representation of the work one by a constant force 7 3 W Thus far we ture vary w how far the

28 F Work Done by a Variable Force We can x 2 also apply this iea when F (x) is not constant. O Position sition We can approximate the area uner the curve by breaking it up (a) into rectangles an aing the area of (b) each rectangle. ifferent v the total a the x axis, Force Force n x 2 O x 1 x 2 O Position (b) x 1

29 Work Done by a Variable Force sition (b) the x axis, just as in part (a). The approximation becomes more accurate when we break it up into more rectangles. Force n x 2 O x 1 x 2 Position It we break it up into an infinite number (c) of infinitesimally thin rectangles, we will be evaluating the integral of the force with onstant respect values to that thefollow isplacement the shape of of the curve. object. (b) The

30 Question te the What is the work The one net by work the one force inicate by this force in the graph as the particle movesis from the x area = 0uner to x = 6the m? curve. force reases coul ecause not fit ticle is otion, F x (N) x (m) Figure 7.8 (Example 7.4) The force acting on a particle is constant for the first 4.0 m of motion an then

31 Question te the force reases coul ecause not fit ticle is otion, What is the work The one net by work the one force inicate by this force in the graph as the particle movesis from the x area = 0uner to x = 6the m? curve. W = 25 J. F x (N) x (m) Figure 7.8 (Example 7.4) The force acting on a particle is constant for the first 4.0 m of motion an then

32 ll rectangles, we see that the work one by the Work Done stretching a Spring One important example of a force that varies with an object s isplacement is the spring force. e, from e, we see he x axis. a ifferase is the 12 which, of calcuinicate Equilibrium position of spring Force of spring kx +kx Applie force th a series 8 (a).it ual to the ximation (c).inthe force x = 0 x F app = kx x

33 Work Done stretching a Spring F app = kx 192 CHAPTER 7 WORK AND KINETIC ENERGY Force Area = W kx show that the corresp gin. Therefore, the w the general position 1 area is equal to 2 1bas kx. As a result, the wo neee to compress Work to Stretch or Co W = 1 2 kx2 SI unit: joule, J O x Position FIGURE 7 10 Work neee to stretch We can get a feelin spring in the follow What is the worka one spring a byistance the applie x force in stretching the spring The work one is equal to the shae a istance x? area, which is a right triangle. The area EXERCISE 7 4 of the triangle is 1 2 1x21kx2 = 1 2 kx2. The spring in a pinbal require to compress t

34 Work Done stretching a Spring F app = kx 192 CHAPTER 7 WORK AND KINETIC ENERGY Force Area = W show that the corresp gin. Therefore, the w the general position 1 area is equal to 2 1bas kx. As a result, the wo neee to compress Work to Stretch or Co W = 1 2 kx2 SI unit: joule, J O x Position FIGURE 7 10 Work neee to stretch We can get a feelin spring in the follow What is the worka one spring a byistance the applie x force in stretching the spring The work one is equal to the shae a istance x? area, which is a right triangle. The area EXERCISE 7 4 of the triangle Wis 1 2 1x21kx2 = 1 app = 1 2 kx2. 2 kx 2 The spring in a pinbal require to compress t kx

35 Summary energy work net work Homework Walker Physics: NEW: Ch 7, onwar from page 210. Questions: 1, 3, 5; Problems: 5, 7 NEW: Ch 7, onwar from page 210. Problems: 13, 21, 29, 35