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Physics 11a Potential Energy and Energy Conservation Work Kinetic Energy Work-Kinetic Energy Theorem Gravitational Potential Energy Elastic Potential Energy Work-Energy Theorem Conservative and Non-conservative Forces Conservation of Energy November 3, 2008 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 cos q ) x F is the magnitude of the force Δ x is the magnitude of the object’s displacement q is the angle between F and x November 3, 2008 Work Done by Multiple Forces If more than one force acts on an object, then the total work is equal to the algebraic sum of the work done by the individual forces Wnet Wby individual forces Remember work is a scalar, so this is the algebraic sum Wnet Wg WN WF ( F cos q )r November 3, 2008 Kinetic Energy and Work Kinetic energy associated with the motion of an object 1 2 KE 2 mv Scalar quantity with the same unit as work Work is related to kinetic energy 1 2 1 2 mv mv0 ( Fnet cos q )x 2 2 Wnet KEf KEi KE November 3, 2008 Work done by a Gravitational Force Gravitational Force Magnitude: mg Direction: downwards to the Earth’s center Wnet 1 2 1 2 mv mv0 2 2 Work done by Gravitational Force W F r cosq F r Wg mgr cos q November 3, 2008 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 The gravitational potential energy PE mgy 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) November 3, 2008 Reference Levels 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 November 3, 2008 Work and Gravitational Potential Energy PE = mgy Wg Fy cos q mg ( yi y f ) cos 0 mg ( y f yi ) Units of Potential Energy are the same as those of Work and Kinetic Energy Wgrav ity PEi PE f November 3, 2008 Extended Work-Energy Theorem The work-energy theorem can be extended to include potential energy: Wnet KEf KEi KE Wgrav ity PEi PE f If we only have gravitational force, then Wnet Wgravity KE f KEi PEi PE f KE f PE f PEi KEi The sum of the kinetic energy and the gravitational potential energy remains constant at all time and hence is a conserved quantity November 3, 2008 Extended Work-Energy Theorem We denote the total mechanical energy by E KE PE Since The total mechanical energy is conserved and remains the same at all times KE f PE f PEi KEi 1 2 1 2 mvi mgyi mv f mgy f 2 2 November 3, 2008 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 November 3, 2008 Platform Diver A diver of mass m drops from a board 10.0 m above the water’s surface. Neglect air resistance. (a) Find is speed 5.0 m above the water surface (b) Find his speed as he hits the water November 3, 2008 Platform Diver (a) Find is speed 5.0 m above the water surface 1 1 mvi2 mgyi mv2f mgy f 2 2 1 0 gyi v 2f mgy f 2 v f 2 g ( yi y f ) 2(9.8m / s 2 )(10m 5m) 9.9m / s (b) Find his speed as he hits the water 1 0 mgyi mv 2f 0 2 v f 2 gyi 14m / s November 3, 2008 Spring Force Involves the spring constant, k Hooke’s Law gives the force F kd F is in the opposite direction of x, always back towards the equilibrium point. k depends on how the spring was formed, the material it is made from, thickness of the wire, etc. Unit: N/m. November 3, 2008 Potential Energy in a Spring Elastic Potential Energy: 1 2 PEs kx 2 SI unit: Joule (J) related to the work required to compress a spring from its equilibrium position to some final, arbitrary, position x Work done by the spring 1 2 1 2 Ws ( kx)dx kxi kx f xi 2 2 Ws PEsi PEsf xf November 3, 2008 Extended Work-Energy Theorem The work-energy theorem can be extended to include potential energy: Wnet KEf KEi KE Wgrav ity PEi PE f Ws PEsi PEsf If we include gravitational force and spring force, then Wnet Wgravity Ws ( KE f KEi ) ( PE f PEi ) ( PEsf PEsi ) 0 KE f PE f PEsf PEi KEi KEsi November 3, 2008 Extended Work-Energy Theorem We denote the total mechanical energy by E KE PE PEs Since The total mechanical energy is conserved and remains the same at all times ( KE PE PEs ) f ( KE PE PEs )i 1 2 1 1 1 mvi mgyi kxi2 mv2f mgy f kx2f 2 2 2 2 November 3, 2008 A block projected up a 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? November 3, 2008 A block projected up a incline Point A (initial state): vi 0, yi 0, xi 10cm 0.1m v f 0, y f h d sin q , x f 0 Point B (final state): d 1 2 1 2 1 2 1 2 mvi mgyi kxi mv f mgy f kx f 2 2 2 2 1 2 kxi mgy f mgd sin q 2 1 2 2 kxi mg sin q 0.5(625 N / m)( 0.1m) 2 (0.5kg)(9.8m / s 2 ) sin 30 1.28m November 3, 2008 A block projected up a incline Point A (initial state): vi 0, yi 0, xi 10cm 0.1m v f ?, y f h / 2 d sin q / 2, x f 0 Point B (final state): 1 2 1 1 1 mvi mgyi kxi2 mv2f mgy f kx2f 2 2 2 2 1 2 1 2 h k 2 kxi mv f mg ( ) xi v 2f gh 2 2 2 m h d sin q (1.28m) sin 30 0.64m k 2 vf xi gh m ...... 2.5m / s November 3, 2008 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 … November 3, 2008 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 Wg PEi PE f mgyi mgy f Work done by gravity Work done by spring force 1 2 1 2 Ws PEsi PEsf kxi kx f 2 2 November 3, 2008 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 Wnc F d f k d Wotherforces It is generally dissipative. The dispersal of energy takes the form of heat or sound November 3, 2008 Extended Work-Energy Theorem The work-energy theorem can be written as: Wnet KEf KEi KE Wnet Wnc Wc Wnc represents the work done by nonconservative forces Wc represents the work done by conservative forces Any work done by conservative forces can be accounted for by changes in potential energy W PE PE c i f Wg PEi PE f mgyi mgy f Gravity work Spring force work 1 2 1 2 Ws PEi PE f kxi kx f 2 2 November 3, 2008 Extended Work-Energy Theorem Any work done by conservative forces can be accounted for by changes in potential energy Wc PEi PE f ( PE f PEi ) PE Wnc KE PE ( KE f KEi ) ( PE f PEi ) Wnc ( KE f PE f ) ( KEi PEi ) Mechanical energy include kinetic and potential energy 1 2 1 2 E KE PE KE PEg PEs mv mgy kx 2 2 Wnc E f Ei November 3, 2008 Problem-Solving Strategy Define the system to see if it includes non-conservative forces (especially friction, drag force …) Without non-conservative forces 1 2 1 1 1 mv f mgy f kx2f mvi2 mgyi kxi2 2 2 2 2 With non-conservative forces W ( KE PE ) ( KE PE ) nc f f i i 1 1 1 1 fd Wotherforces ( mv2f mgy f kx2f ) ( mvi2 mgyi kxi2 ) 2 2 2 2 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 November 3, 2008 Conservation of Mechanical Energy A block of mass m = 0.40 kg slides across a horizontal frictionless counter with a speed of v = 0.50 m/s. It runs into and compresses a spring of spring constant k = 750 N/m. When the block is momentarily stopped by the spring, by what distance d is the spring compressed? Wnc ( KE f PE f ) ( KEi PEi ) 1 2 1 1 1 mv f mgy f kx2f mvi2 mgyi kxi2 2 2 2 2 1 2 1 2 0 0 kd mv 0 0 2 2 1 1 0 0 kd 2 mv 2 0 0 2 2 d m 2 v 1.15cm k November 3, 2008 Changes in Mechanical Energy for 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 stats from rest at the top. The surface friction can be negligible. Use energy methods to determine the speed of the crate at the bottom of the ramp. 1 1 1 1 fd Wotherforces ( mv2f mgy f kx2f ) ( mvi2 mgyi kxi2 ) 2 2 2 2 1 1 1 1 ( mv2f mgy f kx2f ) ( mvi2 mgyi kxi2 ) N 2 2 2 2 d 1m, yi d sin 30 0.5m, vi 0 y f 0, v f ? 1 ( mv2f 0 0) (0 mgyi 0) 2 v f 2 gyi 3.1m / s November 3, 2008 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 stats 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. 1 1 1 1 fd Wotherforces ( mv2f mgy f kx2f ) ( mvi2 mgyi kxi2 ) 2 2 2 2 N 1 2 k Nd 0 ( mv f 0 0) (0 mgyi 0) 2 fk k 0.15, d 1m, yi d sin 30 0.5m, N ? N mg cos q 0 k dmg cos q 1 2 mv f mgyi 2 v f 2 g ( yi k d cos q ) 2.7m / s November 3, 2008 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 stats from rest at the top. The surface in contact have a coefficient of kinetic friction of 0.15. How far does the crate slide on the horizontal floor if it continues to experience a friction force. 1 1 1 1 fd Wotherforces ( mv2f mgy f kx2f ) ( mvi2 mgyi kxi2 ) 2 2 2 2 1 k Nx 0 (0 0 0) ( mvi2 0 0) 2 k 0.15, vi 2.7m / s, N ? N mg 0 1 k mgx mvi2 2 2 v x i 2.5m 2 k g November 3, 2008 Block-Spring Collision A block having a mass of 0.8 kg is given an initial velocity vA = 1.2 m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Assuming the surface to be frictionless, calculate the maximum compression of the spring after the collision. 1 2 1 1 1 mv f mgy f kx2f mvi2 mgyi kxi2 2 2 2 2 1 2 1 2 mvmax 0 0 mvA 0 0 2 2 xmax m 0.8kg vA (1.2m / s) 0.15m k 50 N / m November 3, 2008 Block-Spring Collision A block having a mass of 0.8 kg is given an initial velocity vA = 1.2 m/s to the right and collides with a spring whose mass is negligible and whose force constant is k = 50 N/m as shown in figure. Suppose a constant force of kinetic friction acts between the block and the surface, with µk = 0.5, what is the maximum compression xc in the spring. 1 1 1 1 fd Wotherforces ( mv2f mgy f kx2f ) ( mvi2 mgyi kxi2 ) 2 2 2 2 1 1 k Nd 0 (0 0 kxc2 ) ( mvA2 0 0) 2 2 N mg and d xc 1 2 1 2 kxc mvA k mgxc 2 2 25 xc2 3.9 xc 0.58 0 xc 0.093m November 3, 2008 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 November 3, 2008 Conservation of Energy Energy is conserved This means that energy cannot be created nor destroyed 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 November 3, 2008 Practical Case E=K+U =0 The total amount of energy in the system is constant. 1 2 1 1 1 mv f mgy f kx2f mvi2 mgyi kxi2 2 2 2 2 November 3, 2008 Practical Case K + U + Eint = W + Q + TMW + TMT + TET + TER The Work-Kinetic Energy theorem is a special case of Conservation of Energy K + U = W November 3, 2008 Ways to Transfer Energy Into or Out of A System Work – transfers by applying a force and causing a displacement of the point of application of the force Mechanical Waves – allow a disturbance to propagate through a medium Heat – is driven by a temperature difference between two regions in space Matter Transfer – matter physically crosses the boundary of the system, carrying energy with it Electrical Transmission – transfer is by electric current Electromagnetic Radiation – energy is transferred by electromagnetic waves November 3, 2008 Connected Blocks in Motion Two blocks are connected by a light string that passes over a frictionless pulley. The block of mass m1 lies on a horizontal surface and is connected to a spring of force constant k. The system is released from rest when the spring is unstretched. If the hanging block of mass m2 fall a distance h before coming to rest, calculate the coefficient of kinetic friction between the block of mass m1 and the surface. fd Wotherforces KE PE 1 PE PEg PEs (0 m2 gh) ( kx2 0) 2 1 2 k Nx 0 m2 gh kx 2 N mg and xh 1 k m1 gh m2 gh kh2 2 1 m2 g kh 2 k m1 g November 3, 2008 Power Work does not depend on time interval The rate at which energy is transferred is important in the design and use of practical device The time rate of energy transfer is called power The average power is given by W P t when the method of energy transfer is work November 3, 2008 Instantaneous Power Power is the time rate of energy transfer. Power is valid for any means of energy transfer Other expression W Fx P t Fv t A more general definition of instantaneous power W dW dr P lim t 0 t dt F dt F v P F v Fv cosq November 3, 2008 Units of Power The SI unit of power is called the watt 1 watt = 1 joule / second = 1 kg . m2 / s3 A unit of power in the US Customary system is horsepower 1 hp = 550 ft . lb/s = 746 W Units of power can also be used to express units of work or energy 1 kWh = (1000 W)(3600 s) = 3.6 x106 J November 3, 2008 Power Delivered by an Elevator Motor A 1000-kg elevator carries a maximum load of 800 kg. A constant frictional force of 4000 N retards its motion upward. What minimum power must the motor deliver to lift the fully loaded elevator at a constant speed of 3 m/s? Fnet , y may T f Mg 0 T f Mg 2.16 104 N P Fv (2.16 104 N )(3m / s) 6.48 104W P 64.8kW 86.9hp November 3, 2008