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Short Version : 7. Conservation of Energy. 7.1. Conservative & Non-conservative Forces. F is conservative if. for every closed path C. W BA + W AB = 0. W AB = W BA. = W AB. W AB. i.e.,. is path-independent. W BA. W AB.
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7.1. Conservative & Non-conservative Forces F is conservative if for every closed path C. WBA+WAB = 0 WAB = WBA = WAB WAB i.e., is path-independent WBA WAB F is non-conservative if there is a closed path C such that Mathematica is path-dependent
Example: Work done on climber by gravity Going up: W1 = ( m g ) h = m g h Going down: W2 = ( m g ) ( h) = m g h Round trip: W = W1 + W2 = 0 Horizontal displacement requires no work. Gravity is conservative.
Example: Work done on trunk by friction Going right: W1 = ( m g ) L = m g L Going left: W2 = ( m g ) ( L) = m g L Round trip: W = W1 + W2 = 2 m g L 0 Friction is non-conservative.
GOT IT? 7.1. If it takes the same amount of work to push a trunk across a rough floor as it does to lift a weight to the same distance straight upward. How do the amounts of work compare if the trunk & weight are moved along curved paths between the same starting & end points? Ans. Work is greater for the trunk.
7.2. Potential Energy Conservative force: Potential energy = stored work = ( work done by force ) Note: only difference of potential energy matters. 1-D case: Constant F:
Gravitational Potential Energy Horizontal component of path does not contribute. Vertical lift: m g
Elastic Potential Energy x0 = equilibrium position Ideal spring: Let parabolic U is always positive Setting x0 = 0 : x x0 x = x0 x x0
7.3. Conservation of Mechanical Energy Mechanical energy: Law of Conservation of Mechanical Energy: ( no non-conservative forces ) if
Example 7.5. Spring & Gravity A 50-g block is placed against a spring at the bottom of a frictionless slope. The spring has k = 140 N/m and is compressed 11 cm. When the block is released, how high up the slope does it rise? Initial state: Final state:
Example 7.6. Sliding Block A block of mass m is launched from a spring of constant k that is compressed a distance x0. The block then slides on a horizontal surface of frictional coefficient . How far does the block slide before coming to rest? Initial state: Launch: Work done against friction: Final state: Conservation of energy :
7.4. Potential Energy Curves Frictionless roller-coaster track How fast must a car be coasting at point A if it’s to reach point D? turning points Criterion: potential barrier potential well
Example 7.7. H2 Near the bottom of the potential well of H2, U = U0 + a ( x x0 )2 , where U0= 0.760 aJ, a = 286 aJ / nm2 , x0 = 0.0741 nm. ( 1 aJ = 1018 J ) What range of atomic separation is allowed if the total energy is 0.717 aJ? Turning points:
Force & Potential Energy Force ~ slope of potential curve ( x along direction of F )
Gaussian Gun 1 2 Video 1 1 2 Assume fields of theinduced dipolesnegligible compared to that of the magnet.