Transcript Lectures 22
Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lectures 22, 23, 24 Work-energy theorem: total 1 2 W W con 1 2 W nc 1 2 U 2 U1 W nc 1 2 nc 1 2 W nc 1 2 If W KE2 KE1 KE2 KE1 KE2 U 2 KE1 U1 0, KE2 U 2 KE1 U1 const Mechanical energy is conserved! Examples Strategy: write down the total mechanical energy, E, E = KE + U at the initial and final positions of a particle: Initial E1=KE1+U1… Final E2=KE2+U2 Then use nc 1 2 If W or 0, KE2 U 2 KE1 U1 H Water Slide Who hits the bottom with a faster speed? The curve of fastest descent Cycloid – the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line. Inverted cycloid: Brachistochrone http://curvebank.calstatela.edu/brach/brach.htm 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? Conservative Forces If there are only conservative forces in the problem, then there is conservation of mechanical energy • Conservative: Can go back and forth along any path and the potential energy and kinetic energy keep turning into one another – Good examples: Gravity and Springs • Non-Conservative: As you move along a path, the potential energy or kinetic energy is turned into heat, light, sound etc… Mechanical energy is lost. – Good example: Friction (like on Roller Coasters) Law of Conservation of Energy • Mechanical Energy NOT always conserved • If you’ve ever watched a roller coaster, you see that the friction turns the energy into heating the rails, sparks, noise, wind etc. • Energy = Kinetic Energy + Potential Energy + Heat + Others… –Total Energy is what is conserved! K1+U1 = K2+U2+EHeat… Total Energy is what is conserved! K1+U1= K2+U2+EHeat… Force of gravity: F 0i mg j Potential energy function: U mgy Const Spring: F kxi 0 j 2 kx U Const Potential energy function: 2 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 will it go? Quiz A block of mass m is placed against a vertical spring, spring constant k. The spring is unstretched at y=0. A If the spring is compressed an amount A and the block released from rest, how high will it go? Quiz A block of mass m is attached to a vertical spring, spring constant k. The spring is unstretched at y=0. A If the spring is compressed an amount A and the block released from rest, how high will it go? Block of mass m has a 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. H y=0 ENERGY DIAGRAMS Potential Energy Diagrams • For Conservative forces can draw energy diagrams • Equilibrium points – Motion will move “around” the equilibrium – If placed there with no energy, will just stay (no force) Fx dU dx 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? A particle moves along the x-axis while acted on by a single conservative force parallel to the xaxis. The force corresponds to the potentialenergy function graphed in the Figure. The particle is released from rest at point A. a)What is the direction of the force on the particle when it is at point A? b) At point B? c) At what value of x is the kinetic energy of the particle a maximum? d) What is the force on the particle when it is at point C? e) What is the largest value of x reached by the particle during its motion? f) What value or values of x correspond to points of stable equilibrium? g) Of unstable equilibrium? 2 or 3D cases: dU If F dr or 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 )] 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 : Given the potential energy function U ( x, y ) Gm1m2 1 2 2 (x y ) 2 find the x and y components of the corresponding force. Have a great day! Reading: Chapter 9 Hw: Chapter 9 problems and exercises