普通物理 General Physics 8 – Conservation of Energy 郭艷光Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件: ykuo@cc.ncue.edu.tw 網頁: http://ykuo.ncue.edu.tw

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 8-1 What Is Physics? 8-2 Work and Potential Energy 8-3 Path Independence of Conservative Forces 8-4 Determining Potential Energy Values 8-5 Conservation of Mechanical Energy 8-6 Reading a Potential Energy Curve 8-7 Work Done on a System by an External Force 8-8 Conservation of Energy 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 8-1 What Is Physics? As was done in Chapter 7, we use scalars such as work, kinetic energy, and mechanical energy rather than vectors. Therefore the approach is mathematically simpler. Physics determines how the potential energy of a system can be calculated so that energy might be stored or put to use. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-2 Work and Potential Energy Let us throw a tomato upward, the change in gravitational potential energy is defined as being equal to the negative of the work done on the tomato by the gravitational force. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-2 Work and Potential Energy In Fig. (a), spring force does negative work on the block, transferring energy from the kinetic energy of the block to the elastic potential energy of the spring– block system. In Fig. (b), the transfer of energy is reversed. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-2 Work and Potential Energy Conservative and non-conservative forces (1) The work done by conservative forces depends only on the initial and final positions, not on the path taken. ( Ex: gravitational force ) (2) A force that is not conservative is called a nonconservative force. ( Ex: spring force ) 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-3 Path Independence of Conservative Forces The work done by a conservative force around any closed path is zero. If only a conservative force acts on the particle, then the work done on the particle is the same along the two paths. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-3 Path Independence of Conservative Forces Proof of Equation: if the force is conservative, then the net work done during the round trip must be zero And 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-1 A 2.0 kg block of slippery cheese that slides along a frictionless track from point a to point b. The cheese travels through a total distance of 2.0 m along the track, and a net vertical distance of 0.80 m. How much work is done on the cheese by the gravitational force during the slide? 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-1 Solution: along the horizontal segment, the angle is a constant 90°. along the vertical part of the dashed path 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-4 Determining Potential Energy Values The potential energy of a system is the external work needed to bring the particles from the U = 0 configuration to the given positions at constant speed. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-4 Determining Potential Energy Values Gravitational Potential Energy: The gravitational potential energy associated with a particle-Earth system depends only on the vertical position y ( or height ) of the particle relative to the reference position y = 0, not on the horizontal position. ( gravitational potential energy ) 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-4 Determining Potential Energy Values Proof of Equation: 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-4 Determining Potential Energy Values Elastic Potential Energy: ( elastic potential energy ) 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-2 (a) A 2.0 kg sloth hangs 5.0 m above the ground. What is the gravitational potential energy U of the sloth– Earth system if we take the reference point y = 0 to be 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-2 (a) (1) at the ground, (2) at a floor that is 3.0 m above the ground, (3) at the limb, and (4) 1.0 m above the limb? Take the gravitational potential energy to be zero at y 0. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-1 (a) Solution: (1) (2) (3) (4) 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-1 (b) The sloth drops to the ground. For each choice of reference point, what is the change in the potential energy of the sloth–Earth system due to the fall? Solution: for all four situations, we have the same thus, 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-5 Conservation of Mechanical Energy The mechanical energy Emec of a system is the sum of its potential energy U and the kinetic energy K of the objects within it In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum cannot change. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-5 Conservation of Mechanical Energy and This result is called the principle of conservation of mechanical energy. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-5 Conservation of Mechanical Energy When the mechanical energy of a system is conserved, we can relate the sum of kinetic energy and potential energy at one instant to that at another instant without considering the intermediate motion and without finding the work done by the forces involved. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-5 Conservation of Mechanical Energy As a pendulum swings, the energy of the pendulum–Earth system is transferred between kinetic energy K and gravitational potential energy U, with the sum K + U being constant. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-3 A child of mass m is released from rest at the top of a water slide, at height h = 8.5 m above the bottom of the slide. Assuming that the Slide is frictionless because of the water on it, find the child’s speed at the bottom of the slide. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-3 Solution: The conservation of principle tells us 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-4 A ball is projected from the top of a cliff of height H with an initial speed v0 at some angle above the horizontal. Discuss its motion in terms of (1) the equations of kinematics, and (2) the conversation of mechanical energy. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-4 Solution: (1) We use the equation of kinematics with and (2) With Ug = 0 at the base of the cliff and v as the final speed, one setting , we find 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-5 (a) Two blocks with masses m1 = 2 kg and m2 = 3 kg hang on either side of a pulley. Block m1 is on an incline ( = 30° ) and is attached to a spring whose constant is 40 N/m. The system is released from rest with the spring at its natural 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-5 (a) length. Find the maximum extension of the spring. ( ignore friction and the pulley. ) Solution: 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-5 (b) The speed of m2 when the extension is 0.5 m. ( ignore friction and the pulley. ) Solution: and inserting the given values we find 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-6 Reading a Potential Energy Curve A conservative force can be derived from a scalar potential energy function. solving for F(x) and passing to the differential limit yield 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-6 Reading a Potential Energy Curve The Potential Energy Curve: At x2 , x3 and x4, the slope of the U(x) vs. x curve is zero, thus F = 0. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-6 Reading a Potential Energy Curve The slope dU/dx between x3 and x4 is negative; Thus F > 0 for the this interval. The slope dU/dx between x2 and x3 is positive; Thus F < 0 for the this interval. We can easily find F(x) by (graphically) taking the slope of the U(x) curve at various points. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-6 Reading a Potential Energy Curve Turning Points: : Motion is allowed 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-6 Reading a Potential Energy Curve The points at which: are known as turning points for motion at the turning points . : Motion is forbidden 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-6 Reading a Potential Energy Curve Minima in the U versus x curve are positions of stable equilibrium. Maxima in the U versus x curve are positions of unstable equilibrium. A position at which the slope dU/dx = 0 and thus F = 0 is called an equilibrium point. x > x5: neutral equilibrium 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-6 (a) A 2.00 kg particle moves along an x axis in one- dimensional motion while a conservative force along that axis acts on it. The potential energy U(x) associated with the force is plotted in Fig. (a). That is, if the particle were placed at any position between x = 0 and x = 7.00 m, it would have 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-6 (a) the plotted value of U. At x = 6.5 m, the particle has velocity . From Fig. (a), determine the particle’s speed at x1 = 4.5 m. Solution: At x = 6.5 m, the particle has kinetic energy 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-6 (a) At x = 4.5 m, the potential energy is U1 = 7.0 J. The kinetic energy K1 is the difference between Emec and U1: 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-6 (b) Where is the particle’s turning point located? Solution: Which gives us d = 2.08 m 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-6 (c) Evaluate the force acting on the particle when it is in the region 1.9 m < x < 4.0 m. Solution: For the graph of Fig. (b), we see that for the range 1.0 m < x < 4.0 m the force is 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-7 Work Done on a System by an External Force Work is energy transferred to or from a system by means of an external force acting on that system. No friction involved: Friction involved: Emec: thermal energy by sliding 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-7 The prehistoric people of Easter Island carved hundreds of gigantic stone statues in a quarry and then moved them to sites all over the island (Figure). How they managed to move the statues by as much as 10 km without the use of sophisticated machines has been hotly debated. They most likely cradled each statue in a wooden sled and 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-7 then pulled the sled over a “runway” consisting of almost identical logs acting as rollers. In a modern reenactment of this technique, 25 men were able to move a 9000 kg Easter Island-type statue 45 m over level ground in 2 min. (1) Estimate the work the net force from 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-7 the men did during the 45 displacement of the statue, and determine the system on which that force did the work. (2) What was the increase in the thermal energy of the system during the 45 m displacement? (3) Estimate the work that would have been done by the 25 men if they had moved the statue 10 km across level ground on Easter 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-7 Island. Also estimate the total change that would have occurred in the statue– sled–logs–ground system. Solution: 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-7 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-8 Conservation of Energy Conservation of Energy: The total energy E of a system can change only by amounts of energy that are transferred to or from the system. Isolated system: The total energy E of an isolated system cannot change. 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

8-8 Conservation of Energy Power: The average power due to the force is The instantaneous power due to the 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-8 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 = 10000 N/m, by what distance d is the spring compressed when 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-8 the package stops? Solution: 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-9 Figure shows the mountain slope and the valley along which a rock avalanche moves. The rocks have a total mass m, fall from a height y = H, move a distance d1 along a slope of angle , and then move a distance d2 along a flat valley. What is the ratio d2/H of the runout to the fall height if the coefficient of kinetic friction has the reasonable value of 0.60? 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 Example 8-9 Solution: 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 8! 2018/11/22 普通物理講義-8 / 國立彰化師範大學物理系/ 郭艷光教授