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Physics 111: Mechanics Lecture 7 Bin Chen NJIT Physics Department Chapter 7 Potential Energy and Energy Conservation 7.1 Gravitational Potential Energy 7.2 Elastic (Spring) Potential Energy 7.3 Conservative and Non-conservative Forces 7.4* Force and Potential Energy 7.5* Energy Diagrams Definition of Work W The work, W, done by a constant force on an object is defined as the product of the component of the force along the direction of displacement and the magnitude of the displacement 𝑊 =𝑭∙𝒔 = 𝐹𝑠 cos𝜃 F is the magnitude of the force s is the magnitude of the object’s displacement q is the angle between F and s andDFx Work by gravitation force A 10-kg cannonball was fired from ground toward a target located in an enemy compound that is 20 m away and 15 m high. How much work is done by the gravitational force on the cannonball as it just hits the target? A. 1.5 kJ; B. -1.5 kJ; C. 2.0 kJ; D. -2.0 KJ; E. 3.2 kJ Wgrav mgy1 mgy2 Work done by Gravity and Gravitational Potential Energy U mgy (gravitational potential energy) Wgrav Ugrav rav grav Fs w y1 y2 Wgrav Ugrav,1 Ugrav,2 mgy mgy1 mgy2 Ugrav,2 Ugrav,1 ( Ugrav (gravitational potential energy) K1 Ugrav,1 K2 Ugrav,2 (if only gravity does work) Ugrav,1 Ugrav,2 Ugrav,2 Ugrav,1 Ugrav ( 1 1 2 Ugrav,1 2 Fs w y1 y2 m mgy1 2 1 1 2 m 2 2 K2 Ugrav,2 Uel mgy1 1 2 m 1 2 2 2 1 mgy2 (if only gravity does work) (if only gravity does work) kx2 ( (elastic potential energy) mgy2 (if only gravity does work) 2 ( 1 2 ( Potential Energy Potential energy is associated with the position of the object Gravitational Potential Energy is the energy associated with the relative position of an object in space near the Earth’s surface Shared by both the object and Earth The gravitational potential energy m is the mass of an object g is the acceleration of gravity y is the vertical position of the mass relative the surface of the Earth SI unit: joule (J) Reference Level A location where the gravitational potential energy is zero must be chosen for each problem The choice is arbitrary since the change in the potential energy is the important quantity Choose a convenient location for the zero reference height often the Earth’s surface may be some other point suggested by the problem Once the position is chosen, it must remain fixed for the entire problem Reference Levels Choose the correct answer. The gravitational potential energy of an object (a) is always positive (b) is always negative (c) never equals to zero (d) can be negative or positive W Fscos (constant force, straight-line d Work-Kinetic W FEnergy s (constantTheorem force, straight-line disp When work is done by a net force on an object 2 1 m (definition of kinetic ene and the only changeKin the object is its speed, 2 the work done is equal to the change in the object’s kinetic energy The work done by the net force on a particle equals the chan Speed will increase if work is positive kinetic energy: Speed will decrease if work is negative Wtot W x2 x1 K2 K1 K (work energy theorem 1 1 2 𝑊tot = 𝑚𝑣2 − 𝑚𝑣12 2 2 Fx dx (varying x-component of force, straight- K y Equations 1 2 m 2 (definition of kinetic energ Extended Work-Energy Theorem with Wgrav Fs w y1 y2 mgy1 mgy2 Potential Energy The workGravitational done by the net force on a particle equals the chang Theenergy: work-kinetic energy to kinetic Wgrav Fs theorem w y1 can y2 be extended mgy1 mgy 2 include gravitational potential energy: Ugrav mgy (gravitational potential energy) Wtot K2 K1 K (work energy theorem) Wgrav Ugrav U grav,1 mgy Ugrav,2 (gravitational potential energy) Ugrav,2 Ugrav,1 Ugrav x2 W only have Fx dx gravitational (varying x-component of all force, straight-lin If we force and work x1 W Ugrav,2are(ifzero, Ugrav,2 Ugrav,1 by allU then grav,1 Kdone U grav Krest Uforces only gravity does work) U grav 1 grav,1 2 grav,2 𝑊𝑡𝑜𝑡 = 𝑊𝑔𝑟𝑎𝑣 P2 P2 P2 W F cos 2 dl F||dl F d (work done on 1 K 1K P1 P1 (If only Pgravity U U (if only gravity does work 1 does work) m mgy m mgy (if only gravity does work) grav,1 2 grav,2 1 2 2 2 2 2 11 1 m 2U mgykx el 1 2 21 m W energy) P potential (average mgy (if only gravitypower) does work 2 (elastic grav 1 Wgrav 2 1 Fs w y1 y2 2 mgy1 mgy2 (gravitational potential energy) ConservationU of mgy Mechanical Energy grav Ugrav mgy (gravitational potential energy We denote the total mechanical energy by Wgrav U grav,1 Ugrav,2 Ugrav,2 W U U grav,2 K1gravUgrav,1 grav,1 K2 Ugrav,2 Since So Ugrav,1 U Ugrav U grav,1 (if grav,2 only gravity does work 2 2 K112 mU1grav,1 only gravity mgy1K2 12 mUgrav,2 mgy2 (if(if only gravity doesdoes work 2 The total mechanical energy is conserved and remains the same at all times 2 1 U kx 2 2 (elastic potential energy) 1 el 21 2 m 1 mgy1 2 m 2 mgy2 (if only gravity does (If only gravity does work) 2 1 1 Wel 2 kx1 2 kx22 Uel,1 Uel,2 Uel 1 2 kx 2 Uel (elastic potential energy) Problem-Solving Strategy Define the system Select the location of zero gravitational potential energy Do not change this location while solving the problem Identify two points the object of interest moves between One point should be where information is given The other point should be where you want to find out something Platform Diver A diver of mass m drops from a board 10.0 m above the water’s surface. Neglect air resistance. Find his speed as he hits the water Platform Diver Find his speed as he hits the water 1 2 1 mvi mgyi mv2f mgy f 2 2 1 2 0 mgyi mv f 0 2 v f 2 gyi 14m / s Skateboarding A boy skateboards from rest down a curved frictionless ramp. He moves through a quarter-circle with radius R=3m. The boy and his skateboard have a total mass of 25 kg. Find his speed at the bottom of the ramp Skateboarding Find his speed at the bottom of the ramp K1 0 U grav,1 mgR K 2 12 mv22 U grav, 2 0 0 mgR 1 2 mv2 0 2 1 2 1 mv1 mgy1 mv22 mgy2 2 2 v2 2gR 7.67m / s Prof. Walter Lewin’s Pendulum https://www.youtube.com/watch?v=xXXF 2C-vrQE Conservation of Mechanical Energy Three identical balls are thrown from the top of a building, all with the same initial speed. The first ball is thrown horizontally, the second at some angle above the horizontal, and the third at some angle below the horizontal. Neglecting air resistance, rank the speeds of the balls as they reach the ground, from fastest to slowest. (a) 1, 2, 3 (b) 2, 1, 3 (c) 3, 1 ,2 (d) 3, 2, 1 (e) all three ball strike the ground at the same speed K pter 7 Key Equations 2 m (definition of kinetic The work done by the netGravity force on a particle equals the When Forces other than Do Work W Fs w y y mgy mgy 1 2 1 2 kinetic grav energy: The work-kinetic energy theorem can be extended to include potential energy: Wtot K2 potential K1 K (work energy the U mgy (gravitational energy) grav 𝑊𝑡𝑜𝑡 = 𝑊𝑔𝑟𝑎𝑣 + 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝐾2 − 𝐾1 Since Then Wgrav x2 U grav,1W Ugrav,2Fx dx U(varying Ugrav,1 Ugravof force, stra x-component grav,2 x1 𝑊𝑜𝑡ℎ𝑒𝑟 + 𝑈𝑔𝑟𝑎𝑣,1 − 𝑈𝑔𝑟𝑎𝑣,2 = 𝐾2 − 𝐾1 K1 Ugrav,1 K2 Ugrav,2P2 (if only gravity does P2 P2 work) Fdo coswork dl F||dl F d (work do If forces other than W gravity P1 P1 P1 1 2 𝐾1 2+ 𝑈 𝑊2 𝑜𝑡ℎ𝑒𝑟 =(if𝐾only 𝑔𝑟𝑎𝑣,11 + 2+𝑈 𝑔𝑟𝑎𝑣,2does work) m mgy m mgy gravity 1 1 2 2 2 W 1 1 𝑚𝑣12 + 𝑚𝑔𝑦11 +2 𝑊𝑜𝑡ℎ𝑒𝑟 = 𝑚𝑣22 P +av 𝑚𝑔𝑦t2 Uel 2 kx (elastic 2potential energy) 2 Wel 1 2 kx12 1 2 kx22 UPel,1 lim Uel,2 W dW Uel (average power (instantaneous Lifting bucket from well A person lifted a 10.0-kg bucket from the bottom of a 10-m deep well. The bucket started from rest and had a 1.00 m/s speed as it reached the top. How much work, in J, is done by this person? (g=9.8 m/s2. Neglect friction.) A. B. C. D. E. 975 985 980 -985 -975 Spring Force: An Elastic Force Hooke’s Law gives the force 𝐹𝑥 = 𝑘𝑥 Where 𝐹𝑥 is the force exerted on the spring in the same direction of x The force exerted by the spring is 𝐹𝑠 = −𝐹𝑥 = −𝑘𝑥 k is the spring constant. Unit: N/m. Work done on the spring from x1 to x2 x2 x2 x1 x1 W Fx dx kxdx 1 2 1 2 kx2 kx1 2 2 Potential Energy in a Spring Work done by the spring 𝑥2 𝑊𝑒𝑙 = 𝑥1 1 2 1 2 −𝑘𝑥 𝑑𝑥 = 𝑘𝑥1 − 𝑘𝑥2 2 2 Elastic Potential Energy: SI unit: Joule (J) related to the work required to compress a spring from its equilibrium position to some final, arbitrary, position m 1 2 1 2 m mgy1 2 1 1 2 K1 Ugrav,1 m 2 2 mgy2 (if only gravity does K2 UKgrav,212 m 2 (definition only gravity doesenergy) work) of kinetic Work-Energy(ifTheorem with Extended m 12 mgy1 12 m 22 mgy2 (if only gravity does work) Elastic Potential Energy 2 1 U kx (elastic potential energy) 2 2 1 1 2 force The work done by the net on a particle change el m mgy m mgy (if only equals gravitythe does work)in the p 1 1 2 2 2 2 kinetic The work-kinetic energy theorem can be extended to energy: 2 1 Uel elastic (elastic potential energy) include energy: 2 kx potential K22 2K1(elastic (work energy theorem) 1 K 2potential UelWtot 12 1kx energy) Wel 1 2 kx12 1 2 x2 2 kx1 2 kx2 kx22 Uel,1 Uel,2 Uel,1 Uel,2 Uel Uel 2 2 1 1 W have F dx (varying x -component of force, straight-line displa W kx kx Uand U U If we only spring force all done x 1 2 el,1 el,2work el 2 2 x1 el by all forces zero, then 𝑊the=elastic Krest Kare force does 𝑊 1 Uel,1 2 Uel,2 (if only K1 Uel,1 2 1 Wel 𝑒𝑙 K2 Uel,2 (if only the elastic force𝑡𝑜𝑡 does work) P2 only the P2 elastic force does work) K1 Uel,1P2 K2 Uel,2 (if (If the elastic W F cos dl F||dl F only d (work done on a curved P1 P1 P1 force does work) 2 2 1 1 2 2 1 12 1 m kx 11 kx 2 2 mv kx 22 1 1 2 2 2 2 12 2 22 1 2 m 1 K1 U1 kx22 (if only the elastic force does work) W Pav (average power) Wother K2 U2 (validt in general) K in general) K1 UU1 WWotherK K U2 U2(valid in(valid general) 1 2 kx1 elastic force the doeselastic work) force mv2(if only kxthe (if only 2 2 1 2 mv22 1 2 Hitting a Spring A 2 kg block slides with no friction and with an initial speed of 4 m/s. It hits a spring with spring constant k=1400 N/m. The block compresses the spring in a straight line for a distance 0.1 m. What is the block’s speed, in m/s, at that point? A. B. C. D. E. 1.0 2.0 3.0 4.0 5.0 K1 Ugrav,1 K2 Ugrav,2 (if only gravity does work) K 1 2 m 2 ( (definition of kinetic energy) Extended Work-Energy Theorem 2 2 1 1 m mgy m mgy2 (ifand only Elastic gravity does work) Energy ( with BOTH Gravitational Potential 1 1 2 2 2 The work done by the net force on a particle equals the change in the The work-kinetic energy theorem can be extended to kinetic energy: include both of potential energy: 2 1 types Uel 2 kx (elastic potential energy) Wtot K2 K1 K (work energy theorem) ( 𝑊𝑔𝑟𝑎𝑣 = 𝑚𝑔𝑦1 − 𝑚𝑔𝑦2 = 𝑈𝑔𝑟𝑎𝑣,1 − 𝑈𝑔𝑟𝑎𝑣,2 = −∆𝑈𝑔𝑟𝑎𝑣 Wel 12 kx1x2 12 kx22 Uel,1 Uel,2 Uel (7 W Fx dx (varying x-component of force, straight-line displ x If the system only involves gravitational force and spring force or all work done by all rest forces are zero, Kthen K2 U (if only the+Pelastic force does work) (7 1 Uel,1 P 𝑊Pel,2 = 𝑊 𝑊 𝑡𝑜𝑡 𝑔𝑟𝑎𝑣 𝑒𝑙 2 1 W 1 2 m 2 P1 F cos dl 2 P1 2 F||dl P1 F d (work done on a curved 𝐾1 + 𝑈𝑔𝑟𝑎𝑣,1 + 𝑈𝑒𝑙,1 = 𝐾2 + 𝑈𝑔𝑟𝑎𝑣,2 + 𝑈𝑒𝑙,2 2 1 1 2 kx12 1 12 mv22 1 2 kx22 1 (if only 1 the elastic force 1 does work) 𝑚𝑣12 + 𝑚𝑔𝑦1 + 𝑘𝑥12 = 𝑚𝑣22 +W𝑚𝑔𝑦2 + 𝑘𝑥22 (average power) 2 2 2Pav 2 K1 U1 Wother K2 U2 (7 t (valid in general) W dW (7 Mechanical Energy Conservation with BOTH Gravitational and Elastic Potential Energy We denote the total mechanical energy by Since 𝐸2 = 𝐸1 The total mechanical energy is conserved and remains the same at all times 1 1 2 1 1 2 2 2 𝑚𝑣1 + 𝑚𝑔𝑦1 + 𝑘𝑥1 = 𝑚𝑣2 + 𝑚𝑔𝑦2 + 𝑘𝑥2 2 2 2 2 A block projected up an incline A 0.5-kg block rests on a horizontal, frictionless surface. The block is pressed back against a spring having a constant of k = 625 N/m, compressing the spring by 10.0 cm to point A. Then the block is released. (a) Find the maximum distance d the block travels up the frictionless incline if θ = 30°. (b) How fast is the block going when halfway to its maximum height? A block projected up a incline Point A (initial state): Point C (final state): A block projected up a incline Point A (initial state): Point B (final state): Wel 1 2 kx12 1 2 kx22 Uel,1 Uel,2 Uel If other forces do work K U K U (if only the elastic force does wo 1 el,1 2 el,2 If work done by other forces that cannot be described in terms of 2potential 2 energy 2 2 1 1 1 1 2 m 1 2 kx1 2 mv2 2 kx2 (if only the elastic force do General case If Wother is positive, E = K + U increases K U E =UKint+ U0 decreases (law of conservation of energy) If Wother is negative, If Wother is zero, E = K + U keeps constant Fx x K1 U1 Wother dU x dx K2 U2 (valid in general) (force from potential energy, one dime Types of Forces Conservative forces Work and energy associated with the force can be recovered Examples: Gravity, Spring Force, EM forces Nonconservative forces The forces are generally dissipative and work done against it cannot easily be recovered Examples: Kinetic friction, air drag forces, normal forces, tension forces, applied forces … March 4, 2015 Conservative Forces A force is conservative if the work it does on an object moving between two points is independent of the path the objects take between the points The work depends only upon the initial and final positions of the object Any conservative force can have a potential energy function associated with it Work done by gravity Work done by spring force Nonconservative Forces A force is nonconservative if the work it does on an object depends on the path taken by the object between its final and starting points. The work depends upon the movement path For a non-conservative force, potential energy can NOT be defined Work done by a nonconservative force It is generally dissipative. The dispersal of energy takes the form of heat or sound Conservation of Energy in General Any work done by conservative forces can be accounted for by changes in potential energy 𝑊𝑐 = 𝑈1 − 𝑈2 = − 𝑈2 − 𝑈1 = −∆𝑈 Law of conservation of energy Energy is never created or destroyed. It only changes form. Problem-Solving Strategy Define the system to see if it includes non-conservative forces (especially friction, drag force …) Without non-conservative forces With non-conservative forces Select the location of zero potential energy Do not change this location while solving the problem Identify two points the object of interest moves between One point should be where information is given The other point should be where you want to find out something Homework #7 review I: Conservative Forces A child's toy consists of a block that attaches to a table with a suction cup, a spring connected to that block, a ball, and a launching ramp. The spring has a spring constant k, the ball has a mass m, and the ramp rises a height y above the table, the surface of which is a height H above the floor. Initially, the spring rests at its equilibrium length. The spring then is compressed a distance s, where the ball is held at rest. The ball is then released, launching it up the ramp. When the ball leaves the launching ramp its velocity vector makes an angle θ with respect to the horizontal. ignore friction and air resistance. Calculate vr, the speed of the ball when it leaves the launching ramp. Express the speed of the ball in terms of k, s, m, g, y, and/or H. With what speed will the ball hit the floor? Express the speed in terms of k, s, m, g, y, and/or H. Work-Energy Example for NonConservative Forces A 5-N force pushes an 2-kg block across a surface a distance 2 m. The block starts with a speed of 4 m/s and ends with 3 m/s. How much work, in J, does the friction do? A. -10 B. -3 C. -17 D. 3 E. 9 Another Example: Sliding down a slope A crate slides down a slope with an angle of inclination of 30 degrees. The coefficient of moving friction between the crate and the slope is 0.15. What is the speed of the crate after it N moves 1 m along the slope? A. B. C. D. E. 2.7 m/s 3.1 m/s 3.5 m/s 1.2 m/s Can’t tell, need mass fk Changes in Mechanical Energy for Non-conservative forces A 3-kg crate slides down a ramp. The ramp is 1 m in length and inclined at an angle of 30° as shown. The crate starts from rest at the top. The surface in contact have a coefficient of kinetic friction of 0.15. Use energy methods to determine the speed of the crate at the bottom of the ramp. N fk Energy Review Kinetic Energy Associated with movement of members of a system Potential Determined by the configuration of the system Gravitational and Elastic Internal Energy Energy Related to the temperature of the system Energy is conserved Energy cannot be created nor destroyed It can be transferred from one object to another or change in form If the total amount of energy in a system changes, it can only be due to the fact that energy has crossed the boundary of the system by some method of energy transfer