普通物理 General Physics 7 – Work-Kinetic Energy Theorem 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-7 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 7-1 What Is Physics? 7-2 What Is Energy? 7-3 Kinetic Energy 7-4 Work 7-5 Work and Kinetic Energy 7-6 Work Done by the Gravitational Force 7-7 Work Done by a Spring Force 7-8 Work Done by a General Variable Force 7-9 Power 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 7-1 What Is Physics? We will develop the work-kinetic energy theorem and apply it to solve a variety of problems. This approach is alternative approach to mechanics. It uses scalars such as work and kinetic energy rather than vectors such as velocity and acceleration. 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 7-2 What Is Energy? Technically, 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. 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 7-3 Kinetic Energy Kinetic Energy (K): K is energy associated with the state of motion of an object. For an object of mass m whose speed v is well below the speed of light, The SI unit of kinetic energy (and every other type of energy) is the joule (J). (kinetic energy) 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-1 In 1896 in Waco, Texas, William Crush parked two locomotives at opposite ends of a 6.4-km-long track, fired them up, tied their throttles open, and then allowed them to crash head-on at full speed in front of 30,000 spectators. Hundreds of people were hurt by flying debris; several were killed. Assuming each locomotive weighed and its acceleration was a constant 0.26 m/s2, 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-1 what was the total kinetic energy of the two locomotives just before the collision? 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-1 Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 7-4 Work We account for these changes in K by saying that F has transferred energy W to or from the object. If energy is transferred to m (its K increases), we say that work was done by F on the object (W > 0). If on the other hand energy is transferred from the object (its K decreases), we say that work was done by m (W < 0). 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-5 Work and Kinetic Energy Finding an expression for Work: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-5 Work and Kinetic Energy We use only the force component along the object’s displacement to calculate the work. we can also write The force component perpendicular to the displacement does zero work. (work done by a constant force) (work done by a constant force) 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-5 Work and Kinetic Energy There are two restrictions to calculate work done on an object by a force. (1) The force must be a “constant force”. (2) The object must be “particle-like”. Positive work: Negative work: When two or more forces act on an object, the net work done on the object is the sum of the works done by the individual forces. 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-5 Work and Kinetic Energy Work-Kinetic Energy Theorem: which says that 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-2 (a) Figure shows two industrial spies sliding an initially stationary 225 kg floor safe a displacement of magnitude 8.50 m, straight toward their truck. The push of spy 001 is 12.0 N, directed at an angle of 30.0° downward from the horizontal; the pull of spy 002 is 10.0 N, directed at 40.0° above the horizontal. The magnitudes and directions of these forces do not change as 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-2 (a) the safe moves, and the floor and safe make frictionless contact. What is the net work done on the safe by forces and during the displacement ? 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-2 (a) Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-2 (b) During the displacement, what is the work done on the safe by the gravitational force and what is the work done on the safe by the normal force from the floor? Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-2 (c) The safe is initially stationary. What is its speed at the end of the 8.50 m displacement ? Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-3 (a) During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement while a steady wind pushes against the crate with a force . How much work does this force do on the crate during the displacement? 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-3 (a) Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-3 (b) If the crate has a kinetic energy of 10 J at the beginning of displacement , what is its kinetic energy at the end of ? Solution: Using the work-kinetic energy theorem, we have 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-6 Work Done by the Gravitational Force We use mg as the magnitude of . Thus, the work (1) For the rising object, (2) The object is falling back down, 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-6 Work Done by the Gravitational Force Work done by a force in Lifting an object: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-6 Work Done by the Gravitational Force 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-6 Work Done by the Gravitational Force Work done by a force in Lowering an object: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-4 (a) One of the lifts of Paul Anderson in the 1950s remains a record: Anderson stooped beneath a reinforced wood platform, placed his hands on a short stool to brace himself, and then pushed upward on the platform with his back, lifting the platform straight up by 1.0 cm. The platform held automobile parts and a safe filled with lead, with a total weight of 27900 N (6270 lb). As Anderson 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-4 (a) lifted the load, how much work was done on it by the gravitational force . Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-4 (b) How much work was done by the force Anderson applied to make the lift? solution: WA done by Anderson’s applied force was the negative of the work Wg done by the gravitational force . 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-5 (a) An initially stationary 15.0 kg crate of cheese wheels is pulled, via a cable, a distance d = 5.70 m up a frictionless ramp to a height h of 2.50 m, where it stops. How much work is done on the crate by the gravitational force during the lift? 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-5 (a) Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-5 (b) How much work is done on the crate by the force from the cable during the lift? Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-6 A skier of mass m = 40 kg is given a displacement of 20 m along a slop inclined at to the horizontal. The tension in the towrope is T = 250 N and acts at an angle α = 30° to the incline. Given that , find the work done by each force and the net work on the skier. 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-6 Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-6 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-7 (a) An elevator cab of mass m = 500 kg is descending with speed when its supporting cable begins to slip, allowing it to fall with constant acceleration During the fall through a distance d = 12 m, what is 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-7 (a) the work done on the cab by the gravitational force ? Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-7 (b) During the 12 m fall, what is the work done on the cab by the upward pull of the elevator cable? Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-7 (b) 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-7 (c) What is the net work W done on the cab during the fall? Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-7 (d) What is the cab’s kinetic energy at the end of the 12 m fall? Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-7 Work Done by a Spring Force Hooke’s Law : The spring force is given by The minus sign indicates that the direction of the spring force is always opposite the direction of the displacement. k is called the spring constant. (Hooke’s law) 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-7 Work Done by a Spring Force The Work Done by a Spring Force: two assumptions about the spring: (1) Block’s mass is massless. (2) Spring is an ideal spring. 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-7 Work Done by a Spring Force Show that the expression Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-8 (a) A package of spicy Cajun pralines lies on a frictionless floor, attached to the free end of a spring in the arrangement of Fig. (a). A rightward applied force of magnitude Fa = 4.9 N would be needed to hold the package at x1 = 12 mm. How much work does the spring force do on the package if the package is pulled rightward from x0 = 0 to x2 = 17 mm? 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-8 (a) Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-8 (b) Next, the package is moved leftward to x3 = 12 mm. How much work does the spring force do on the package during this displacement? Explain the sign of this work. Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-9 A cumin canister of mass m = 0.40 kg slides across a horizontal frictionless counter with speed v = 0.50 m/s. It then runs into and compresses a spring of spring constant k = 750 N/m. When the canister is momentarily stopped by the spring, by what distance d is the spring compressed? 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-9 Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-8 Work Done by a General Variable Force One-Dimensional Analysis: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

7-8 Work Done by a General Variable Force Work-Kinetic Energy Theorem with a Variable Force: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-10 In an epidural procedure, the feel a doctor has for the needle’s penetration is the variable force that must be applied to advance the needle through the tissues. Figure is a graph of the force magnitude F versus displacement x of the needle tip in a typical epidural procedure. (The line segments have been straightened somewhat from the original data.) As x increases from 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-10 from 0, the skin resists the needle, but at x = 8.0 mm the force is finally great enough to pierce the skin, and then the required force decreases. Similarly, the needle finally pierces the interspinous ligament at x = 18 mm and the relatively tough ligamentum flavum at x = 30 mm. The needle then enters the epidural space (where it is to deliver the anesthetic fluid), and the force drops sharply. 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-10 Anew doctor must learn this pattern of force versus displacement to recognize when to stop pushing on the needle. Thus this is the pattern to be programmed into a virtual- reality simulation of an epidural procedure. How much work W is done by the force exerted on the needle to get the needle to the epidural space at x = 30 mm? 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-10 Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-10 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-11 Force , with x in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates ( 2 m, 3 m ) to ( 3 m, 0 m )? Does the speed of the particle increase, decrease, or remain the same? 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-11 Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 7-9 Power Power: The time rate at which work is done by a force. Average power: Instantaneous power: ( SI unit of power is Watt ) 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-12 Figure shows constant forces and acting on a box as the box slides rightward across a frictionless floor. Force is horizontal, with magnitude 2.0 N; force is angled upward by 60° to the floor and has magnitude 4.0 N. the speed v of the box at a certain instant is 3.0 m/s. what is the power due to each force acting on the box at that 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-12 instant, and what is the net power? Is the net power changing at that instant? solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-13 Provided a funny car does not lose traction, the time it takes to race from rest through a distance D depends primarily on the engine’s power P. assuming the power is constant, derive the time in term of D and P. 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 Example 7-13 Solution: 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 7! 2018/12/6 普通物理講義-7 / 國立彰化師範大學物理系/ 郭艷光教授