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Chapter 7 Potential Energy and Energy Conservation

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7.1 Gravitational Potential Energy Conservation of Mechanical Energy (Gravitational Forces Only) Example 7.1 Height of a baseball from energy conservation You throw a 0.145-kg baseball straight up in the air, giving it an initial upward velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance.

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When forces other than gravity do work Example 7.2 Work and energy in throwing a baseball In Example 7.1, suppose your hand moves up 0.50 m while you are throwing the ball, which leaves your hand with an upward velocity of 20.0 m/s. Again ignore 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.

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Gravitational Potential Energy for Motion Along a Curved Path Conceptual Example 7.3 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 neglected.

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7.2 Elastic Potential Energy Situations with both gravitational and elastic potential energy Example 7.9 Motion with gravitational, elastic and friction forces In a “worst-case” design scenario, a 2000-kg elevator with broken cables is 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 the force constant of the spring should be.

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Test Your Understanding of Section 7.2 Consider the situation in Example 7.9 at the instant when the elevator is still moving downward and the spring is compressed by 1.00 m. The kinetic energy K, gravitational potential energy U grav, and elastic potential energy U el at this instant, are they positive or negative?

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7.3 Conservative and Nonconservative Forces Example 7.11 Conservative or nonconservative? In a certain region of space the force on an electron is, where C is a positive constant. The electron moves in a counter-clockwise direction around a square loop in the xy-plane (Fig. 7.20). The corners of the square are at (x,y) = ( 0,0 ), (L, 0 ), (L,L), and ( 0,L). Calculate the work done on the electron by the force during one complete trip around the square. Is this force conservative or nonconservative?

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The Law of Conservation of Energy Test Your Understanding of Section 7.3 In a hydroelectric generating station, falling water is used to drive turbines (water wheels), which in turn run electric generators. Compared to the amount of gravitational potential energy released by the falling water, how much electrical energy is produced? (i) the same; (ii) more; (iii) less.

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7.4 Force and Potential Energy Example 7.13 A 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 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 charged particle, as a function of its position.

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Force and Potential Energy in Three Dimensions Example 7.14 Force and potential energy in two dimensions A puck slides on a level, frictionless air-hocky table. The coordinates of the puck are x and y. It is acted on by a conservative force described by the potential- energy function 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.

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Test Your Understanding of Section 7.4 A particle moving along the x-axis is acted on by a conservative force F x. At a cer point, the force is zero. (a) Which of the following statements about the value of potential-energy function U(x) at that point is correct? (i) U(x)= 0 ; (ii) U(x)> 0 ; (iii) U(x) 0 ; (iii) dU(x)/dx< 0 ; (iv) not enough information is given to decide.

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7.5 Energy Diagrams Test Your Understanding of Section 7.5 The curve in Fig. 7.24b has a maximum at a point between x2 and x3. Which statement correctly describes what happens to the particle when it is at this point? (i) The particle’s acceleration is zero. (ii) The particle accelerates in the positive x-direction; the magnitude of the acceleration is less than at any other point between x2 and x3. (iii) The particle accelerates in the positive x- direction; the magnitude of the acceleration is greater than any other point between x2 and x3. (iv) The particle accelerates in negative x-direction; the magnitude of the acceleration is less than any other point between x2 and x3. (v) The particle accelerates in the negative x-direction; the magnitude of the acceleration is greater than any other point between x2 and x3.

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