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Chapter 9 Work and Energy 第九章 功與能 What is energy? The term energy is so broad that a clear definition is difficult to write. Energy is a scalar quantity associated with the state (or condition) of one or more objects. Energy can be transformed from one type to another and transferred from one object to another, but the total amount is always the same (energy is conserved). No exception to this principle of energy conservation has ever been found. Kinetic energy Before collision: 1-D collision: After collision: m1v1 m1v1 m2v2 Momentum conservation: After collision in the rest frame of particle 2: v1 v1 v2 Thus: v1 m1 m2 v1 m1 m2 v2 2m1 v1 m1 m2 Verify: m1v12 m2v22 m1v12 so the kinetic energy is conserved. 1 2 K .E. mv 2 Why the prefactor should be ½ ? What is work? Energy can be transformed or transferred by force. But force alone might not change the energy of an object, e.g. centripetal force on an object in uniform circular motion. Work W is energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work. Work by a constant force 1 2 K .E. mv 2 Why the prefactor should be ½ ? d d 1 2 ( K .E.) ( mv ) mva Fv dt dt 2 i f t d ( K .E.) Fvdt Fx 0 ( K .E.) Fx work Work done in curvilinear motion dW F dr dv 1 dW F dr m (v dt ) mv dv d ( mv v ) dt 2 1 2 1 2 W dW F dr mv f mvi 2 2 path path Work done by a variable force rf W F rn F dr n ri For 1D along x axis: W xf xi Fx dx Work-energy theorem 1 2 1 2 W mv f mvi 2 2 or W K The unit of energy: 1 J 1 kg m s 1 N m 1 cal 4.186 J 2 -2 Power dW F dr P F v dt dt The unit of power: 1 W 1J s 1 N m s -1 1 horsepower 745.7 W 1 kilowatt hour 3.6 106 J -1 Tables of energy and power Work done by gravity F mg mgj W mg s (mgj ) (xi yj ) W mg ( y f yi ) For arbitrary path: yf W dW mg ds (mg ) dy mg ( y f yi ) yi Work done by a spring Fspring kx xf Wspring kxdx 1 2 2 Wspring k ( x f xi ) 2 xi Work done by friction Wby A on B Wby B on A W f fk s Note that the work done by friction depends on the total path length, not just the initial and final coordinates. fk Example 0 A crate of mass m is dropped onto a conveyor belt that moves at a constant speed v (see the figure below). The coefficient of kinetic friction is k. (a) What is the work done by friction? (b) How far does the crate move before reaching its final speed? (c) When the crate reaches its final speed, how far has the belt moved? Potential energy Potential energy Dutch Scientist Christian Huygens showed that speed v reached of a fall from a height h is given by v2 h Gottfried W. Leibnitz thought a falling body acquires a “force” that can carry it back to its original level. His “force” is what we call energy. Potential energy Potential energy Work-energy theorem leads to the principle of conservation of mechanical energy. Wext U U f U i The zero of the potential energy can be chosen arbitrarily. If the potential energy of the initial configuration is chosen to be zero, i.e. Ui = 0, then the potential energy of a system is the external work that needed to take the system to its final state. Conservative forces If the work done by a force to an object depends on only its initial and final positions and not the path taken, then it is called a conservative force, otherwise it is a nonconservative force. Work done by gravity: Wg mg ( y f yi ) Work done by a spring: 1 Wsp k ( x 2f xi2 ) 2 Work done by friction: Depending on the path. Conservative forces For a conservative force: 1) ( 2) Wa( W b a b Wa(2)b Wb(2)a 1) ( 2) Wa( W b ba 0 The work done by a conservative force around any closed loop is zero. Potential energy and conservative forces Since the work done by a conservative force to an object depends only on its initial and final positions, we can write: WC (U (rf ) U (ri )) The reason for the minus sign will become clear later. If we assume K = 0 when the object is moved, then it requires an external force to counter balance the conservative force FC. It seems more natural to have: WEXT WC U (rf ) U (ri ) Potential energy and conservative forces For a fixed initial position r0, we can rewrite WC as a function of the final position r. WC (r ) U (r ) U (r0 ) dU dWC FC dr U B U A FC dr B A A conservative force can be associated with a scalar potential energy function, whereas a nonconservative force cannot. Gravitational potential energy Gravitational potential energy function (near the Earth surface): U g mgy const The arbitrary constant allows us to choose the zero potential energy at any convenient position. Conservation of mechanical energy If a particle is only subject to a conservative force, then we can use the potential energy with the work-energy theorem, W = K, to obtain: K U 0 K f Ki U f U i 0 K f U f Ki U i We define the mechanical energy E as E K U The principle of the conservation of mechanical energy: E E f Ei 0 Applications of the energy conservation 1 2 E mv mgy contant 2 Example 1 You throw a 0.145-kg baseball up in the air, giving it an initial upward velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance. v12 2 gy1 v22 2 gy2 Work-energy theorem including potential energy Wother WC K 2 K1 WC (U 2 U1 ) Wother ( K 2 U 2 ) ( K1 U1 ) Wother E2 E1 Example 2 In example 1, suppose your hand moves up while you are throwing the ball, which leaves your hand with an upward velocity of 20.0 m/s. Again, ignore the air resistance. (a) Assuming that your hand exerts a constant upward force on the ball, find the magnitude of that force. (b) Find the speed of the ball at a point 15.0 m above the point where it leaves your hand. Motion along a curved path W Ftotal s (w Fother ) s Fother s 0 W w s mg( y2 y1 ) We can use the same expression for gravitational potential energy whether the body’s path is curved or straight. s Example 3 A batter hits two identical baseballs with the same initial speed and height but different initial angles. Prove that at a given height h, both balls have the same speed if air resistance can be ignored. Example 4 Your cousin Throckmorton skateboards down a curved playground ramp. If we treat Throcky and his skateboard as a particle, he moves through a quarter-circle with radius R = 3.0 m. He starts from rest and there is no friction. (a) Find his speed at the bottom of the ramp. (b) Find the normal force that acts on him at the bottom of the curve. 1 2 2 E mv mgy contant v 2 gR v 7.67 m/s 2 2 mv N mg 3mg 735 N R Example 5 Assume the same initial condition as in Example 4 except that the ramp is not frictionless. If Throcky’s speed at the bottom is only 6.00 m/s, then what was the work done by the frictional force acting on him? W friction WC K 2 K1 Example 6 We want to load a 12-kg crate into a truck by sliding it up a ramp 2.5 m long, inclined at 30°. A worker, giving no thought to friction, calculates that he can get the crate up the ramp by giving it an initial speed of 5.0 m/s at the bottom and letting it go. But friction is not negligible; crate slides 1.6 m up the ramp, stops, and slides back down. (a) Assume that the friction force acting on the crate is constant, find it magnitude. (b) How fast the crate moving when it reaches the bottom of the ramp? Elastic potential energy Work done by the spring: Wel x2 x1 1 (kx)dx k ( x22 x12 ) 2 Spring potential energy function: 1 2 U el kx 2 1 2 1 2 E mv kx contant 2 2 Example 7 A glider with mass m = 0.200 kg sits on a frictionless horizontal air track, connected to a spring with force constant k = 5.00 N/m. You pull on the glider, stretching the spring 0.100 m, and then release it with no initial velocity. The glider begins to move back toward its equilibrium position (x = 0). What is its x-velocity when x = 0.080 m? Example 8 For the system of Example 7, suppose the glider is initially at rest at x = 0, with the spring unstretched. You then apply a constant force F in the +x-direction with magnitude 0.610 N to the glider. What is the glider’s velocity when it has moved to x = 0.100 m? Problem solving guide for energy conservation 1. In general there may be more than one type of potential energy and more than one particle kinetic energy involved in the problem. E K U g U sp 2. Care must be taken with the signs and the zero of the potential, which should not be changed in one problem once it is fixed. Example 9 In a worst case design scenario, a 2000-kg elevator with broken cables falling at 4.00 m/s when it first contacts a cushioning spring at the bottom of the shaft. The spring is supposed to stop the elevator, compressing 2.00 m as it does so. During the motion a safety clamp applies a constant 17,000-N frictional force to the elevator. As a design consultant, you are asked to determine what force constant of spring should be?. Nonconservative forces A force that is not conservative is called a nonconservative force. The work done by a nonconservative force: 1. cannot be expressed as the difference of a potential-energy function. 2. is not reversible. 3. is path-dependent. 4. is not zero when ending point is the same as the starting point. Some nonconservative forces, such as kinetic friction or fluid resistance, always cause the mechanical energy to be dissipated. They are called dissipative forces. There are nonconservative forces that increase mechanical energy, e.g. exploding firecracker Example 10 You are rearranging your furniture and wish to move a 40.0-kg futon 2.50 m across the room. However, the straight-line path is block by a heavy coffee table that you don’t want to move. Instead, you slide the futon in a dogleg path over the floor; the doglegs are 2.00 m and 1.50 m long. Compared to the straight-line path, how much work must you do to push the futon in the dogleg path? The coefficient of kinetic friction is 0.200. Example 11 In a certain region of space the force on an electron is F = Cxj, where C is a positive constant. The electron moves in a counterclockwise direction around a square loop in the xy-plane. The corners of the square are (x, y) = (0, 0), (L, 0), (L, L), (0, L). Calculate the work done on the electron by the force F during one complete trip around the square. Is this force conservative or not? Mechanical energy and nonconservative forces The work-energy theorem: Wnet WC WNC K WC U WNC K U The law of conservation of energy: WNC U int K U U int 0 External force and internal energy transfer An external force can change the kinetic energy or potential energy of an object without doing work on the object—that is, without transferring energy to the object. Instead, the force is responsible for transfers of energy from one type to another inside the object. Example 12 A 2.0 kg package of tamale slides along a floor with speed v1 4.0 m/s. It then runs into and compresses a spring, until the package momentarily stops. Its path to the initially relaxed spring is frictionless, but as it compresses the spring, a kinetic frictional force from the floor, of magnitude 15 N, acts on the package. If k 10 000 N/m, by what distance d is the spring compressed when the package stops? Example 13 A block of mass m is attached to a spring and moves on a rough inclines as in the following figure. Initially, the block is at rest with the spring unextended. A force F acting at an angle α to the incline pulls the block. Write the modified form of the workenergy theorem. Example 14 A block of mass m = 0.2 kg is held against, but not attached to a spring (k = 50 N/m) which is compressed by 20 cm, as shown below. When released, the block slides 50 cm up the rough incline before coming to a rest. Find: (a) the force of friction; (b) the speed of the block as it leaves the spring. Conservative forces and potential energy functions A conservative force can be derived from a scalar potential energy function. Example 15 An electrically charged particle is held at rest at the point x = 0, while a second particle with equal charge is free to move along the positive x-axis. The potential energy of the system is U(x) = C/x, where C is a positive constant that depends on the magnitude of the charges. Derive an expression for the x-component of force acting on the movable particle, as a function of its position. Conservative forces and potential energy functions Force and potential energy function in three dimensions: dU F dr ( Fx dx Fy dy Fz dz ) U U U dU dx dy dz x y z U ˆ U ˆ U ˆ ˆ ˆ ˆ F (i j k ) (i j k )U x y z x y z ˆ ˆ ˆ (i j k ) x y z F U Example 16 A puck slides on a level, frictionless air-hockey table. The coordinates of the puck are x and y. It is acted on by a conservative force described by the potential-energy function U(x, y) = ½ k(x2 + y2). Derive an expression for the force acting on the puck, and find an expression for the magnitude of the force as a function of position. Energy diagrams Note that x = 0 is a stable equilibrium point. Potential energy curve Note that the points x1 and x3 are stable equilibrium points; while the points x2 and x4 are unstable equilibrium points. Energy diagrams K E U A particle is in a bound state when E < 0. Gravitational potential energy GmM F 2 rˆ r Along a particular radial direction r̂ : r2 r2 r1 r1 U (r2 rˆ) U (r1rˆ) F dr GmM GmM U (r2 ) U (r1 ) r2 r1 GmM U (r ) r GmM dr 2 r Generalized conservation of energy Mechanical energy can be dissipated into heat through nonconservative forces, e.g. frictional forces. However, we can include the thermal energy generated in the heating process, and it is possible to convert some of the thermal energy back into mechanical energy. Around 1845, several scientists proposed the principle of the conservation of energy: Energy can change its form, but it can neither be created nor destroyed. Summary The work done by a force: WAB B A The work-energy theorem: W K The instantaneous mechanical power: P F v dE / dt F dr Summary Conservative force: WA(1) B WA(2) B Potential energy: U WC A conservative force can be derived from a scalar potential energy function: U Fx x U Fy y U F U r U Fz z Conservation of mechanical energy: E K U WNC Gravitational potential energy: U GmM r Further reading & homework • The Feynman lectures on Physics, Vol. 1, Chapter 4, 13, and 14. • Questions: 9.1, 9.4, 9.6, 9.11, 9.12, 9.15 • Problems: 9.1, 9.5, 9.9, 9.11, 9.14, 9.19, 9.21, 9.27