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11/30/2014 Resultant Work So far: Motion analysis with forces. NOW: An alternative analysis using the concepts of Work & Energy. This box is moved 4m with the Net Force Resultant work is also equal to the work of the RESULTANT force. 4m 80 N -10 N Easier? Yes Conservation of Energy: NOT a new law! Just Newton’s Laws in a different language. Work = (F - f) x Work = (80 - 10 N)(4 m) Work = 280 N m Box gains 280N m of work Definition of Work Work by a Constant Force Work is a scalar quantity equal to the product of the displacement x and the component of the force Fx in the direction of the displacement. Work = Force component d displacement Work = Fx d Work = (F cos ) x WORK AND ENERGY ARE SCALAR Force must be in direction of displacement If one dimension (Cos 0 = 1) W = F d Work is negative if force is opposite of direction Resultant Work (Net Work) Resultant work is the algebraic sum of the individual works of each force. x Ff Fa Fa = 80 N, Ff = -10 N and x = 4 m I do 80N while Friction does -10 over 4m Net Work = (80 N)(4 m) + (-10 N)(4 m) Work of a Force at an Angle F = 70 N Work = Fx x 60o x = 12 m Work = (F cos ) x Work = (70 N) Cos 600 (12 m) = 420 J Work = 420 N m Only the x-component of the force does work! Net Work on box= 280 Nm 1 11/30/2014 UNIT OF WORK / ENERGY If a bag is pulled angle of 37 deg, d = 50 m, and an of F = 30 N, what is the work done on the bag? Find the work done if m = 50 kg, FP = 1000 N, Ffr = 500 N, θ = 37º Find the work done if m = 50 kg, Ffr = 0 N, θ = 45º, h=20m Work done only depends on force against gravity, could have lifted it and done same work W = Fd = Fdcosθ • Can exert a force & do no work! Could have d = 0 W=0 Or Could have F d θ = 90º, cosθ = 0 W=0 Eight books, each 4.3 cm thick with mass 1.7 kg, lie flat on a table. How much work is required to stack them one on top of another? . 2 11/30/2014 A (frictionless) lawn mower is pushed a horizontal distance of 20 m by a force of 200 N directed at an angle of 300 with the ground. What is the work of this force? x = 20 m F 300 F = 200 N Work is positive since Fx and x are in the same direction. You drag a vacuum cleaner 3.5 m by pulling on the hose – it makes an angle of 30.0 with a frictionless surface. The tension is 55.0 N. What is the work? T 55 N ff n rope makes an angle of 350 with the floor and the work done on the block? By Friction T = 40 N; x = 8 m, uk = 0.2; = 350; m = 4 kg fk mg x A 40-N force pulls a 4-kg Tire a horizontal distance of 8 m. The +x 3.5 m x k n = 0.2. What is T 40 N mg +x 8m WORK AND INCLINE PROBLEMS • OK GO THIS TOO SHALL PASS http://www.youtube.com/watch?v=qybUFnY 7Y8w • HONDA VIDEO 3 11/30/2014 A 330-kg piano slides 3.6 m down a 28º incline and is kept from accelerating by a man who is pushing (Fp) back on it parallel to the incline The effective coefficient of kinetic friction is 0.40. Calculate: (a)the force exerted by the man, (b) the work done by the man on the piano, (c) the work done by the friction force, (d) the work done by (Fp AKA G), and (e) the net work done on the piano. (a) WORK AGAINST A FORCE KE + STORED ENERGY • Chair example lifting const vel against gravity (all work goes into gravity. Then accelerating it against gravity extra work goes into motion • Now push against friction const rate - then push w acceleration (great force (b) (c) The angle between friction and the direction of motion is 180o WORK CLIMING STAIRS (+ WORK – WORK) (d) (e) What is the minimum work (zero a) needed to for a woman to push a 950-kg car 810 m up along a 9.0º incline? (a) Ignore friction. (b) Assume the effective coefficient of friction retarding the car is 0.25. (a) You pull a 55.5 kg wooden box at 250N with a rope that makes a 28.0 angle with the horizontal. The coefficient of kinetic friction between the box and the deck is 0.1. You pull the crate a distance of 2.25 m. How much work was 250 N done? fk mg (b) x n T +x 2.25 m . 4 11/30/2014 You pull a 100 kg wooden box with a rope that makes a 40.0 angle with the horizontal at a constant speed. The coefficient of kinetic friction between the box and the deck is 0.2. You pull the crate a distance of 5 m. How much work was done on the box? Work-Energy Principle Combine all three Net work done on an object by a net force is equal to the change in its Kinetic Energy. Kinetic Energy; Work-Energy Principle • Energy The ability to do work • Kinetic Energy The energy of motion “Kinetic” Greek word for motion An object in motion has the ability to do work Work-Energy Principle • Net work on an object = Change in KE. Wnet = KE – Opposing forces (work) store energy /reduces KE – Wnet is a scalar. – Wnet can be positive or negative – (because KE can be both + & -) – Units are Joules for both work & KE. Kinetic Energy Kinetic Energy: Work-Energy Principle • IF Work >0 • IF Work <0 • IF Work =0 KE INCREASES KE DECREASES KE DOES NOT CHANGE CLIMING STAIRS EXAMPLE: The Work-Energy Principle If displacement is upwards the work by gravity is – the work by you is + If displacement is downwards the work by gravity is + the work by you is - Net work done on an object by a net force is equal to the change in its Kinetic Energy. Do not confuse getting tired with doing work 5 11/30/2014 A net 6500 N force is applied to a resting 1500 kg car, moving it forward. What is its kinetic energy and velocity after 150 m? A circus manager pounds a 1.0 kg tent stake with a wooden mallet that weighs 12.5 Kg at 8 m/s . What is the initial velocity of the stake? since car begins at rest = 975 000 J • Moving hammer can do work on nail! Crider pushes a frictionless tire of mass 5.0 kg across a floor at a rate of 2.0 m/s2 for 7.0 s. Find the net work done on the tire. Use the KE formula to determine the energy in the tire. For hammer: Wh = KEh = -Fd = 0 – (½)mh(vh)2 For nail: Wn = KEn = Fd = (½)mn(vn)2 - 0 A quarterback throws a football (from rest) at 10 m/s. The football weighs 1.5 kilograms. A receiver catches it. How much work/energy was used to accomplish both tasks? How much work must be done to stop a 1250-kg car traveling at 105 km/hr? The work done on the car is equal to the change in its kinetic energy, and so 6 11/30/2014 Mechanical Universe • Lesson 14: Potential Energy • The nature of stability. Potential energy provides a clue, and a powerful model, for understanding why the world has worked the same way since the beginning of time. • Text Assignment: Chapter 14 (Advanced text -- Chapter 10) • Instructional Objectives • Be able to calculate the potential-energy function associated with a given conservative force. • Be able to find the force F(x) from the potential-energy function U(x). • Be able to locate equilibrium points and discuss their stability from a graph of the potential-energy function U(x). • Be able to use the gravitational potential-energy and conservation of energy to solve the problems of escape velocity. Potential Energy (PE or U) Potential Energy: stored energy http://www.youtube.com/watch?v=ujXnhCfrjX8 Net work done by falling object Reference Levels http://www.youtube.com/watch?v=alo_XWCqNUQ Potential Energy Potential Energy • Potential Energy (PE) Energy associated with position or configuration of a mass. • Gravitational Potential Energy: h = distance above Earth m has potential to do work mgh when it falls Y positions are chosen for ease of problem solving: Set the bottom displacement = 0 7 11/30/2014 Gravitational PE Potential energy in the case depends only on vertical displacement Consider a problem in which the height of a mass above the Earth changes from y1 to y2: Change in Gravitational PE is: (PE)grav = mg(y2 - y1) Work done on the mass: W = (PE)grav y = distance above Earth Where we choose y = 0 is arbitrary, since we take the difference in 2 y’s in (PE)grav A 12,500 kg boulder is 157 m above a cabin. What is its PE above the cabin? What will be the KE As the boulder crashes onto the house. What will be the velocity? y=0 Mass is not important, change in velocity is only impacted by h y=0 8 11/30/2014 IF THE Rollercoaster weighs 1000kg find the PE change A 7.0-kg monkey swings from one branch to another 1.2 m higher. What is the change in potential energy? By how much does the gravitational potential energy of a 64-kg pole vaulter change if her center of mass rises about 4.0 m during the jump? • WATCH A VIDEO ON KINETIC POTIENTIAL ENERGY HERE • DO A DEMO (a) Find the force required to give a helicopter of mass M an acceleration of 0.10 g upward. (b) Find the work done by this force as the helicopter moves a distance h upward. A 1.60-m tall person lifts a 2.10-kg book from the ground so it is 2.20 m above the ground. What is the potential energy of the book relative to (a) the ground, and (b) the top of the person’s head? 9 11/30/2014 A 12,500 kg boulder is 157 m above a cabin. What is its PE above the cabin? What will be the KE As the boulder crashes onto the house. What will be the velocity? A 2-kg ball at rest is released from a height of 20 m. What is its velocity when its height has decreased to 5m? 20m v=0 5m 0 Conservation of Mechanical Energy OR E = KE + PE = Constant If the force of Gravity only force in system: v A 3000kg roller coaster starts from rest at a height of 50m and descends down a frictionless track to a nadir that is 10m high before returning upwards. What is the coaster’s velocity at the nadir? Only height differences matter! Horizontal distance doesn’t matter! A flippin’ powerful method of calculation!! h0 = Initial height, v0 = Initial velocity h = Final height, v = Final velocity E1 = E2 PE1 = mgh KE1 = 0 KE + PE = same as at points 1 & 2 KE1 + PE1 = KE2 + PE2 0 + mgh = (½)mv2 + 0 v2 = 2gh The sum remains constant PE2 = 0 KE2 = (½)mv2 10 11/30/2014 MAKE PROBLEMS FROM THIS WEB SITE http://www.physicsclassroom.com/mmedia/energy/ie.cfm MAKE PROBLEMS FROM THIS WEB SITE http://www.physicsclassroom.com/mmedia/energy/hw.cfm On an icy day Mr C. is late for class, starting from rest, he slides down from Tully’s with a large Americano on the 35.0º incline whose vertical height is 50 m. How fast is he going when he reaches Stadium? Physics students try to grab some Tully’s, but can’t walk up the icy walk. They grab a sled and get a running start in the courtyard. They are traveling at 9.0 m/s when they hit the ice. How high (y axis) do they travel? How far up the hill the 35deg incline is that? 11 11/30/2014 http://www.youtube.com/watch?v=_7IRSQ39pDQ&safe=active Sect. 6-5: Conservative Forces WHEN IS MECHANICAL ENERGY IS NOT CONSERVED WHEN IS MECHANICAL ENERGY IS CONSERVED http://www.youtube.com/watch?v=-SbwmooRTBI&feature=related KE CONSERVED IN AN ELASTIC COLLISION BECAUSE ALL OF THE WORK/ENERGY IS PASSED ALONG AND NOT TRANSFERRED INTO HEAT, SOUND, LIGHT, DEFORM ECT… 12 11/30/2014 CAN FRICTION DO POSITIVE WORK? Nonconservative Force - Force leads to dissipation of mechanical energy into another form of energy Conservative Force - Does Not leads to dissipation of mechanical energy Push book on table -- work greater if not in straight line due to friction Friction is nonconservative. Work depends on the path! Energy Conservation • In any process, total energy is neither created nor destroyed. • Energy can be transformed from one form to another & from one body to another, but the total amount remains constant. Law of Conservation of Energy • Again: Not exactly the same as the Principle of Conservation of Mechanical Energy, which holds for conservative forces only! This is a general Law!! • Forms of energy besides mechanical: – Heat (conversion of heat to mech. energy & visaversa) – Chemical, electrical, nuclear, .. Conservative Forces You will be expected to perform mechanics calculations without information about the initial velocity or other data we have used so far, just force Conservative Force The work done by that force depends only on initial & final conditions & not on path taken between the initial & final positions of the mass. PE CAN be defined for conservative forces Non-Conservative Force The work done by that force depends on the path taken between the initial & final positions of the mass. PE CANNOT be defined for non-conservative forces 13 11/30/2014 Conservation of Mechanical Energy For conservative forces ONLY! In any process KE + PE = 0 Conservation of Mechanical Energy E = KE + PE Conservation of mechanical energy In any process,E = 0 = KE + PE E = KE + PE = Constant In any process, the sum of the KE and the PE is unchanged (energy changes form from PE to KE or KE to PE, but the sum remains constant). If several forces act (conservative & non-conservative): The total work done is: Wnet = WC + WNC WC = work done by conservative forces WNC = work done by non-conservative forces The work energy principle still holds: Wnet = KE For conservative forces (by definition of PE): WC = -PE For conservative forces: KE = -PE + WNC OR: WNC = KE + PE Law of Conservation of Energy E(Top)A = E(Bottom)B = E(Top)C = E(Bottom)B = E(Top)D Not the work energy principle since mechanical Energy is not necessarily conserved C A D The total energy is neither decreased nor increased in any process. Energy can be transformed from one form to another & from one body to another, but the total amount IN SYSTEM remains constant. B At D ac must be greater than g At B the velocity is maximum and the KEB = PEA If all forces are conservative PEA = PED & KEA = KED E1 = E2 14 11/30/2014 A 655 N diver leaps into the water from a height of 10.0 m. What is his speed 2.00 m above the water? A bicycle with initial velocity 10 m/s coasts to a net height of 4 m. What is the velocity at the top, neglecting friction? A roller coaster boasts a maximum height of 100.0 m. What is the speed when it reaches its lowest point? A projectile is fired at an upward angle of 45.0º from the top of a 265-m cliff with a speed of 185 m/s. What will be its speed when it strikes the ground below? (Use conservation of energy.) The image part with relationship ID rId6 was not found in the file. In the high jump, Fran’s kinetic energy is transformed into gravitational potential energy without the aid of a pole. With what minimum speed must Fran leave the ground in order to lift her center of mass 2.10 m and cross the bar with a speed of 0.70 m/s? 25.0m Sled and rider together have a mass of 87.0 kg. They are atop a hill elevated at 42.5. They are at a height of 25.0 m (y axis). Find v at bottom of hill. Assume no friction. 15 11/30/2014 Mechanical Universe A 15.0 kg crate slides down a ramp. The ramp is 1.50 m long and is at a 30.0 angle. The crate starts from rest. It experiences a constant frictional force of 7.50 N as it slides downwards. What is the speed of the crate when it reaches the bottom? • Lesson 13: Conservation of Energy • The myth of the energy crisis. According to one of the major laws of physics, energy is neither created nor destroyed. • Text Assignment: Chapter 13 (Advanced text - Chapter 10) • Instructional Objectives • Know the definitions of work, kinetic energy, and potential energy. • Understand the relationship between work and energy. • Be able to work problems using conservation of energy. CONSERVATION OF ENERGY WITH NC WNC = Work done by non-conservative forces KE = Change in KE PE = Change in PE (conservative forces) So, if friction is present, we have (WNC Wfr) Mechanical Universe • Lesson 15: Conservation of Momentum • If The Mechanical Universe is a perpetual clock, what keeps it ticking away till the end of time? Taking a cue from Descartes, momentum -- the product of mass and velocity -- is always conserved. Newton's laws embody the concept of conservation and momentum. This law provides a powerful principle for analyzing collisions, even at the local pool hall. • Text Assignment: Chapter 19 (Advanced text -- Chapter 11) • Instructional Objectives • Recognize conservation of momentum as a consequence of Newton's Second Law. • Know when the momentum of a system is conserved. • Recognize the connection between kinetic energy and momentum. • Be able to solve problems involving elastic and inelastic collisions. • Know the relationship between impulse and time average of force. A 100kg skateboarder comes to a stop (just barely) at the top of a second hill that is 100m from his starting point. What is his force of friction? POWER 16 11/30/2014 Sect. 6-10: Power • Power Rate at which work is done or rate at which energy is transformed: • Average Power: P = (Work)/(Time) = (Energy)/(time) P = W/t • SI units: Joule/Second = Watt (W) 1 W = 1J/s A 1250 kg Elevator carries max load of 955 kg. A constant frictional force of 3850 N exists. What minimum Power for motor is needed to lift the thing at a constant speed of 3.50 m/s? the force is T, so: – British units: Horsepower (hp). 1hp = 746 W “Kilowatt-Hours” (from your power bill). Energy! 1 KWh = (103 Watt) (3600 s) = 3.6 106 W s = 3.6 106 J LAB TASK Average Power: • DETERMINE The s of between your two surfaces using forces Its often convenient to write power in terms of force & speed. A pump is to lift 18.0 kg of water per minute through a height of 3.60 m. What output rating (watts) should the pump have? • DETERMINE k between your two surfaces using energy SPRINGS 17 11/30/2014 There are other types of PE besides gravitational. Calculus Connection Compressed springs store energy x Force by person to compress spring: x spring constant F m K is a measure of how “stiff” the spring is. Force exerted by spring is: Hooke’s Law A 4-kg mass suspended from a spring produces a displacement of 20 cm. What is the spring constant? 20 cm F m Derivative of WS is FS k = 196 N/m The units of k are really kg/s2 Compressing or Stretching a Spring Initially at Rest: WORK Two forces are always present: the outside force Fout ON spring and the reaction force Fs BY the spring. x What work is required to stretch this spring (k = 196 N/m) from x = 0 to x = 30 cm? x m Compressing m Stretching Compression: FMAN does positive work and Fs does negative work Stretching: Fout does positive work and Fs does negative work F 30 cm Note: The work to stretch an additional 30 cm is greater due to a greater average force. 18 11/30/2014 • In a problem in which compression or stretching distance of spring changes from x1 to x2. A 62-kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 12 m, and falls a total of 31 m. Draw a FBD. Calculate the spring stiffness constant k of the bungee cord, assuming Hooke’s law applies. • The change in PE is: (PE)elastic = (½)k(x2)2 - (½)k(x1)2 • The work done: W = - (PE)elastic • The PE belongs to the system, not to individual objects Note: If the height of the platform and the extension height are given you will have PEg for both sides. Conservation of Energy: Energy In A Spring A 0.450 kg silver arrow, resting in a frictionless dart gun is pushed 3.00 cm into a light spring, k = 75.0 N/m. It is then released. What is the velocity of the arrow as it just leaves the spring? Conservation of Energy: this is the PE stored in spring .450 Add to Conservation of mechanical energy equation: A 1200-kg car rolling on a horizontal surface has speed v= 18 m/s when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring stiffness constant of the spring? How far up the frictionless 30o-incline will the 2-kg block move after release? The spring constant is 2000 N/m and it is compressed by 8 cm. Conservation of Energy: Conservation of Energy: 19 11/30/2014 How far up the frictionless 30o-incline will the 2-kg block move after release? The spring constant is 2000 N/m and it is compressed by 8 cm. Sitting on a table that is 100 cm above the deck is a spring that has a spring constant of k = 50 N/m. The spring is compressed a distance of 50 cm with a 0.756 kg ball. (a) What is the kinetic energy stored in the spring? (b) When the spring extends, what is the velocity of the ball? (c) When the ball rolls off the table, how much time does it take till it hits the deck? (d) How far does the ball travel horizontally from the edge of the table before it hits the deck? (e) What is the kinetic energy of the ball just before it hits the deck? (a) Conservation of Energy: (b) (c) (a) A vertical spring (ignore its mass), whose spring stiffness constant is 950 N/m is attached to a table and is compressed down 0.150 m. (a) What upward speed can it give to a 0.30-kg ball when released? (b) How high above its original position (spring compressed) will the ball fly? The level of the ball on the uncompressed spring taken as the zero location for both gravitational PE and elastic PE. So the compressed point It a negative value for gravitational potential. Conservation of Energy: . A ball mass 5.kg falls a distance of 1.2m before hitting a spring. The spring compresses 20cm and stops. Find the v it strikes the spring and solve for k. emember that the compression of the spring is - 0.2m. TARZAN 20 11/30/2014 FORMULAS ON AP Crazy Crider is above the courtyard with a 20.0m rope that makes 40 degrees with the vertical. He swings out on a rope and lets go of the thing when he is at the lowest point of the swing. At this point, he is 15.0 m above the ground. How far Will he travel from the window? EXAM TIME Jane, looking for Tarzan, is running at top speed 5.3 m/s and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward? Does the length of the vine affect your answer? The only forces acting on Jane are gravity and the vine tension. The tension pulls in a centripetal direction, and so can do no work – the tension force is perpendicular at all times to her motion. So Jane’s mechanical energy is conserved. No, the length of the vine does not enter into the calculation, unless the vine is less than 0.7 m long. If that were the case, she could not rise 1.4 m high. Instead she would wrap the vine around the tree branch. 21