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Chapter 7 All forces are CONSERVATIVE or NON-CONSERVATIVE A force is conservative if: The work done by the force in going from r1 to r2 is independent of the path the particle follows or The work done by the force when the particle goes from r1 around a closed path, back to r1 , is zero. Non-conservative: doesn’t satisfy above condition the Theorem: if a force can be written as the gradient (slope) of some scalar function, that force is conservative. 1D case: dU Fx dx U(x) is called the potential energy function for the force F If such a function exists, then the force is conservative x2 dU Fx dx W W con x2 dU Fx dx dx dx x1 x1 [U ( x2 ) U ( x1 )] con does NOT depend on path! If Fx(x) is known, you can find the potential energy function as U ( x) Fx ( x) dx C Work-energy theorem: K 2 K1 W total 1 2 W con 1 2 W nc 1 2 K 2 K1 U 2 U1 W nc 1 2 K 2 U 2 K1 U 1 W nc 1 2 nc 1 2 If W 0, K 2 U 2 K1 U1 const Energy conservation law! A strategy: write down the total energy E = K + U at the initial and final positions of a particle; Then use K 2 U 2 K1 U1 , if W nc 1 2 or K 2 U 2 K1 U 1 W nc 1 2 0 Several dimensions: U(x,y,z) U ( x, y, z ) U ( x, y, z ) U ( x, y, z ) Fx ; Fy ; Fz x y z Partial derivative is taken assuming all other arguments fixed Compact notation using vector del, or nabla: F U , i j k x y z dU Another notation: F dr Geometric meaning of the gradient U Direction of the steepest ascent; Magnitude U : the slope in that direction F U : Direction of the steepest descent Magnitude F : the slope in that direction http://reynolds.asu.edu/topo_gallery/topo_gallery.htm : If or dU F dr U ( x, y, z ) U ( x, y, z ) U ( x, y, z ) Fx ; Fy ; Fz x y z then U ( r2 ) dU W F dr dr dU U (r2 ) U (r1 ) L dr U ( r1 ) W con [U (r2 ) U (r1 )] Work-energy theorem: K 2 K1 W total 1 2 W con 1 2 W nc 1 2 K 2 K1 U 2 U1 W nc 1 2 K 2 U 2 K1 U 1 W nc 1 2 nc 1 2 If W 0, K 2 U 2 K1 U1 const Energy conservation law! Examples y U ( y ) mgy C Force of gravity Fy mg Fx k ( x x0 ) Spring force x k ( x x0 ) U ( x) C 2 2 x0 A block of mass m is attached to a vertical spring, spring constant k. A If the spring is compressed an amount A and the block released from rest, how high will it go from its initial position? A particle is moving in one direction x and its potential energy is given by U(x) = ax2 – bx4 . Determine the force acting on a particle. Find the equilibrium points where a particle can be at rest. Determine whether these points correspond to a stable or unstable equilibrium. Potential Energy Diagrams • For Conservative forces can draw energy diagrams • Equilibrium points – If placed in the equilibrium point with no velocity, will just stay (no force) Fx >0 Fx dU dx a) Spring initially compressed (or stretched) by A and released; b) A block is placed at equilibrium and given initial velocity V0 0 Stable vs. Unstable Equilibrium Points The force is zero at both maxima and minima but… – If I put a ball with no velocity there would it stay? – What if it had a little bit of velocity? Block of mass m has a massless spring connected to the bottom. You release it from a given height H and want to know how close the block will get to the floor. The spring has spring constant k and natural length L. L H y=0 Water Slide Who hits the bottom with a faster speed? H Roller Coaster You are in a roller coaster car of mass M that starts at the top, height H, with an initial speed V0=0. Assume no friction. a) What is the speed at the bottom? b) How high will it go again? c) Would it go as high if there were friction? H Roller Coaster with Friction A roller coaster of mass m starts at rest at height y1 and falls down the path with friction, then back up until it hits height y2 (y1 > y2). Assuming we don’t know anything about the friction or the path, how much work is done by friction on this path? A gun shoots a bullet at angle θ with the x axis with a velocity of magnitude Vm. What is magnitude of the velocity when the bullet returns to the ground? How high it will go? Checking if the force is conservative Fx a; Fy b Fx ax; Fy by Fx ay; Fy bx Fx f ( x); Fy g ( y ) Fx axy; Fy b Checking if U(x,y) is a potential energy for any force 2U ( x, y) 2U ( x, y) xy yx Find the velocity of the block at points 0,B, and D Power Power is a rate at which a force does work If work does not depend on time (or for average power): Otherwise: W P t W Fdr P Fv dt dt Even if instantaneous power depends on time, one can talk about the average power Power could also define the rate at which any form of energy is spent, not only mechanical How many joules of energy does 100 watt light bulb use per hour? How fast would a 70-kg person have to run to have that amount of energy?