Work and Energy • • • • • • Work and kinetic energy total work-kinetic energy theorem Potential energy Mechanic energy Conservation of energy power 1. Work A. Definition In science, the word work has a different meaning than you may be familiar with. The scientific definition of work is: using a force to displace an object (when both the force and the motion of the object are in the same direction.) Work can only be done to a system by an external force; a force from something that is not a part of the system. So if our system is a plane on an aircraft carrier and we come along and push the plane, we can increase the energy of the plane… We are essentially doing work on the plane. B. Calculating Work The amount of work done on an object can be expressed by the given by the equation: 𝑊 = 𝐹 ∙ ∆𝑥 = 𝐹 ∙ ∆𝑥 ∙ 𝑐𝑜𝑠𝜃 Here θ is the angle between force 𝐹 and displacement ∆𝑥 . Meaning, work is the product of the force applied which moves the object a parallel displacement ( we will use “d” for the displacement) W = Fdparallel There are some important points to understand about this equation. Work If the object that is experiencing the force does not move (if d = 0) then no work is done. parallel Negative Work If the object moves in the direction opposite the direction of the force (for instance if force and displacement are in opposite directions) then the work is negative: W < 0. This can happen when a force is applied to slow and object down. Direction of motion F mass Displacement Positive Work If the object moves in the same direction as the direction of the force (for instance if force and displacement are in the same direction) then the work is positive: W > 0. This happens when you push an object to the right to move it rightward. F mass Displacement If there are several forces acting on an object, The total work done by all these forces can be calculated as: 𝑊𝑡𝑜𝑡𝑎𝑙 =𝑊𝐹1 +𝑊𝐹2 + 𝑊𝐹3 +……. 𝑊𝑡𝑜𝑡𝑎𝑙 =𝑊𝑛𝑒𝑡 𝐹𝑜𝑟𝑐𝑒 Units of Work and Energy W =Fd𝒄𝒐𝒔𝜽= Fd parallel This equation gives us the units of work. Since force is measured in Newtons (N) and displacement is measured in meters (m) the unit of work is the Newton-meter (N-m).And since N = kg-m/s ; a N-m also equals a kg-m /s . 2 2 2 However, in honor of James Joule, who made critical contributions in developing the idea of energy, the unit of energy is also know as a Joule (J). J Joule = N-m Newton-meter = kg-m /s 2 2 kilogram-meter2/second2 A +24 N force is applied to an object that moves 10 m in the same direction during the time that the force is applied. How much work is done to the object? answer 1 W= Fd W= (24N)(10m) W= 240J A +24 N force is applied to an object that moves 10 m in the opposite direction during the time that the force is applied. How much work is done to the object? answer 2 W= Fd W= (24N)(-10m) W= -240J A +24 N force is applied to an object that is stationary during the time that the force is applied. How much work is done to the object? answer 3 W= Fd W= (24N)(0m) W= 0 J How much force must be applied to an object such that it gains 100J of energy (which means 100 J of work is done) over a distance of 20 m? answer 4 W= Fd F = W/d F= 100/20 F= 5N Over what distance must a 400 N force be applied to an object such that it gains 1600J of energy (meaning 1600 J of work is done)? answer 5 W= Fd d = W/F d= 1600/400 d= 4m A girl pulls a sled at a constant speed 1.2 m/s by applying a force of 350 N. How much work will be done during 100 seconds? answer 6 d= st d = (1.2)(100) d = 120m W= Fd W= (350)(120) W= 42000J A book is held at a height of 2.0 m for 20 s. How much work is done on the book? answer 7 Object remains stationary therefore no work is done 8 A mg B C D E -mgh mgh 0 -mg Hint: Do a free body diagram to determine a formula for the outside force (Fapp); then use the formula for work: W = Fdparallel. answer A barbell of mass "m" is lifted vertically upwards, at a constant velocity, to a distance "h" by an outside force. How much work does that outside force do on the barbell? C 2. Energy A. Definition It turns out that energy is so fundamental, like space and time, that there is no good answer to this question. However, just like space and time, that doesn't stop us from doing very useful calculations with energy. Energy is defined as the ability to do work. We may not be able to define energy, but because it is a conserved property of nature, it's a very useful idea. 2. Energy B. Types 2. Energy B. Types Energy can come in different forms. 1. Non Mechanical Energy • non mechanical energy is energy that is not related to mechanics; in other words, no work was done • There are five examples you should be familiar with: a. b. c. d. e. Thermal (Heat) Energy Chemical Energy Electrical Energy Nuclear Energy Electromagnetic Radiation (R M I V U X G) 2. Energy B. Types 2. Mechanical Energy • When work is done to an object, it acquires energy.. • The energy it acquires is known as mechanical energy. • There are two forms of mechanical energy: a. Potential Energy – energy of position b. Kinetic Energy – energy of motion 3. Kinetic Energy The energy an object has by virtue of its motion is called its kinetic energy. The symbol we will be using for kinetic energy is KE. Like all forms of energy, it is measured in Joules (J). The amount of KE an object has is given by: KE = ½ mv2 Note: Kinetic energy KE is scalar quantity and will never be negative. As an object falls, its KE always _____. A decreases B increases C stays the same. answer 13 B What is the kinetic energy of a 12 kg object with a velocity of 10 m/s? KE = 0.5 mv E = (0.5)(12)(10 ) 2 K answer 15 KE = 600 J 2 What is the mass of an object which has 2400 J of KE when traveling at 6.0 m/s? KE = 0.5 mv = KE / (0.5)(v ) m = 2400 / (0.5)(6 ) 2 m 2 2 answer 16 m = 133.33kg A 3 kg object has 45 J of kinetic energy. What is its velocity? KE = 0.5 mv = KE/0.5m v = 45 / 0.5 (3) 2 v2 2 answer 17 v = 5.48 m/s If the speed of a car is doubled, the KE of the car is: A quadrupled B quartered C halved D doubled answer 18 A Which graph best represents the relationship between the KE and the velocity of an object accelerating in a straight line? C A KE KE v v B D KE KE v swer 19 v D 20 The data table below lists mass and speed for 4 objects. Which 2 have the same KE? A A and D B and D C A and C D B and C answer B D 4. Potential Energy A barbell of mass "m" is lifted vertically upwards a distance "h" by an outside force. How much work does that outside force do on the barbell? F app mg W = Fdparallel Since a = 0, Fapp = mg W = (mg) dparallel Since F and d are in the same W = (mg) h direction ...and dparallel = h W = mgh 4.1 Gravitational Potential Energy The name for this form of energy—when work is done against gravity--is Gravitational Potential Energy (GPE). GPE = mgh Gravitational Potential Energy, PE=mgh • Energy of position • Stored energy • Symbol: U or PE • Unit: Joule • Compared to a Reference Point (base level) example A flower pot of mass 5kg is located 15m above the ground. A)What is its potential energy with respect to the ground? B) What work was done to raise it to its position? A: 750J What is the change of GPE for a 5.0 kg object which is raised from the floor to a final height of 2.0m above the floor? answer 9 GPE=mgh GPE= (5kg)(9.8)(2m) GPE=98 J 10 As an object falls, its GPE always _____. increases B decreases C stays the same answer A B What is the change of GPE for a 8.0 kg object which is lowered from an initial height of 2.0 m above the floor to a final height of 1.5m above the floor? answer 11 GPE=mgh GPE= (8)(9.8)(-0.5) GPE= -39.2 J What is the change in height of a 2.0 kg object which gained 16 J of GPE? answer 12 GPE=mgh h = GPE/mg h = 16/(2)(9.8) h = 0.82m 4.2. Elastic Potential Energy Energy can be stored in a spring, this energy is called Elastic Potential Energy. Robert Hooke first observed the relationship between the force necessary to compress a spring and how much the spring was compressed. Hooke's Law Fspring = -kx k represents the spring constant and is measured in N/m. x represents how much the spring is compressed and is measured as you would expect, in meters. The - sign tells us that this is a restorative force. (if you let the spring go once it is compressed, it will go back to its original position) Elastic Potential Energy The energy imparted to the spring by this work must be stored in the Elastic Potential Energy (EPE) of the spring: EPE = ½ k x 2 Like all forms of energy, it is measured in Joules (J). Determine the elastic potential energy stored in a spring whose spring constant is 250 N/m and which is compressed 8 cm. EPE = 0.5 kx PE = 0.5 (250)(0.08 ) 2 E answer 21 EPE = 0.08 J 2 What is the spring constant of a spring that is compressed 5 cm and has 0.65 J of elastic potential energy stored in it? EPE = 0.5 kx = EPE/0.5x = 0.65 / 0.5 (0.05 ) 2 k k answer 22 k = 520 N/m 2 2 How much does a spring with a spring constant of 500 N/m need to be compressed in order to store 1.75 J of elastic potential energy? EPE = 0.5 kx = EPE / 0.5k x = 1.75 / 0.5(500) 2 x2 2 answer 23 x = 0.08m A 3 kg mass compresses a spring 2.5 cm. What is the spring constant? answer 24 F = ma F = kx F = 3(9.8) k=F/x F = 29.4N k = 29.4 / 0.025 k = 1176 N/m k = 1176 N/m The same 3 kg mass compresses the same spring 2.5 cm. How much elastic potential energy is stored in the spring? EPE = 0.5kx PE = (0.5)(1176)(0.025 ) 2 E answer 25 EPE = 0.368 J 2 4.4 Mechanical Energy 5. Power 𝑾 𝑷= 𝒕 Average power = Work/Time Power is the rate at which work is done: units: 1 J/s =1 watt (W) Since work is measured in Joules (J) and time is measured in seconds (s) the unit of power is Joules per second (J/s). However, in honor of James Watt, who made critical contributions in developing efficient steam engines, the unit of power is also know as a Watt (W). B. Calculating Since W = Fd parallel Regrouping this becomes Since v = d/t So power can be defined as the product of the force applied and the velocity of the object parallel to that force. A steam engine does 50 J of work in 12 s. What is the power supplied by the engine? answer 26 27 27 A 3.0 kg block is initially at rest on a frictionless, horizontal surface. The block is moved 8.0m in 2.0s by the application of a 12 N horizontal force, as shown in the diagram below. What is the power developed when moving the block? F = 12 N 24 W B 32 W C 48 W D 96 W Frictionless surface 8.0 m answer A 3.0 kg C How long must a 350 W engine run in order to produce 720 kJ of work? answer 28 A 12 kW motor runs a vehicle at a speed of 8 m/s. What is the force supplied by the engine? answer 29 4. Work-Energy Theorem W=KE W=KEf-KEi The net work done on an object is equal to the change in its kinetic energy. Problem If Robin Hood applies a 15N force to pull his bow back by 25cm, with what speed will his 0.150kg arrow leave the bow? A: 7.1m/s Problem 2 What is the work done by gravity as an 8 kg object falls from rest at 15m to 5m? What is the object’s speed at 5m? A: 800J, 14m/s 6. Transformation and Conservation Principles The most powerful concepts in science are called "conservation principles". These principles allow us to solve problems without worrying too much about the details of a process. Conservation Principles A good example is a bag of candy. If you know that there are 50 pieces of candy at the beginning of the day and you know that none of the pieces have been taken out or added...you know that there must be 50 pieces at the end of the day. Conservation Principles Perhaps during the day you changed the way you arrange them by moving them around...but you still will have 50 pieces. In that case we would say that the number of pieces of candy is conserved. That is, we should always get the same amount, regardless of how they are arranged. The same amount exists before and after you moved them around. Conservation Principles We also have to be clear about the system that we're talking about. If we're talking about a specific type of candy...we can't suddenly start talking about a different one and expect to get the same answers. We must define the “system” whenever we use a conservation principle. Conservation of Energy Energy is a conserved property of nature. It is not created or destroyed. Therefore in a closed system we will always have the same amount of energy. The only way the energy of a system can change is if it is open to the outside...this means that energy has been added or taken away. 7. Conservation of Energy • Energy can neither be created nor destroyed. It can only change from form to form. • Mechanical Energy, PE and KE • Conservation of Mechanical Energy PE1 + KE1 = PE2 + KE2 Return to Table of Contents https://www.khanacademy.org/science/physics/work-and-energy/workand-energy-tutorial/v/conservation-of-energy 7.1: Conservative and Nonconservative Forces The gravitational force has an interesting property that when an object is moved from one place to another, the work done by the gravitational force does not depend on the choice of path. Forces like these are called conservative forces. Definition Of A Conservative Force A force is conservative when the work it does on a moving object is independent of the path between the object's initial and final positions. Definition Of A nonconservative Force A force is nonconservative when the work it does on a moving object is dependent of the path between the object's initial and final positions. Examples Conservative Forces Gravitational force (Ch. 4) Elastic spring force (Ch. 10) Electric force (Ch. 18, 19) Nonconservative Forces Static and kinetic frictional forces Air resistance Tension Normal force Propulsion force of a rocket 7.2 The Conservation of Mechanical Energy THE PRINCIPLE OF CONSERVATION OF MECHANICAL ENERGY The total mechanical energy (E = KE + PE) of an object remains constant as the object moves, provided that the net work done by external nonconservative forces is zero. Conservation of Mechanical Energy If friction and wind resistance are ignored, a bobsled run illustrates how kinetic and potential energy can be interconverted, while the total mechanical energy remains constant. A Daredevil Motorcyclist A motorcyclist is trying to leap across the canyon shown in Figure 6.18 by driving horizontally off the cliff at a speed of 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side. Roller Coaster (Ideal) The tallest and fastest roller coaster in the world is now the Steel Dragon in Mie, Japan (Figure 6.20). The ride includes a vertical drop of 93.5 m. The coaster has a speed of 3.0 m/s at the top of the drop. Neglect friction and find the speed of the riders at the bottom. 6.6 Nonconservative Forces and the Work–Energy Theorem In the roller coaster example, we ignored nonconservative forces, such as friction. In reality, however, such forces are present when the roller coaster descends. The actual speed of the riders at the bottom is 41.0 m/s. Assuming again that the coaster has a speed of 3.0 m/s at the top, find the work done by nonconservative forces on a 55.0-kg rider during the descent. Problem 5 A 3kg watermelon sits on a table 1.2m above the ground. a)What is its PE at the top compared to the ground? b) What is its KE at the top? c) With what speed will it hit the ground? A: 36J, 0J, 4.9m/s Problem 6 A penny is at the top of the Empire State Building 381m above the ground. It is then released. With what speed will it hit the ground? A: 87.3 m/s Problem 7-Roller Coaster A roller coaster car started from point A at a height of 100m. What is its speed at point B? A: 44.7m/s Problem 8 - Pendulum a) What is the potential energy of the bob at point A compared to the ground? b) What is the speed of the bob at the bottom? A: 2.3J, 1.1m/s A Conservation of Energy A roller coaster is at the top of a track that is 80 m high. How fast will it be going at the bottom of the hill? Eo + W = Ef Eo = Ef GPE = KE mgh = 0.5mv2 …… solve for v2 V2 = 2gh 2 v = 2 (9.8) 80 v =39.6 m/s answer A spring gun with a spring constant of 250 N/m is compressed 5 cm. How fast will a 0.025 kg dart go when it leaves the gun? answer A student uses a spring (with a spring constant of 180 N/m) to launch a marble vertically into the air. The mass of the marble is 0.004 kg and the spring is compressed 0.03 m. How high will the marble go? h = 2.066m A student uses a spring gun (with a spring constant of 120 N/m) to launch a marble vertically into the air. The mass of the marble is 0.002 kg and the spring is compressed 0.04 m. a)How high will the marble go? b)How fast will it be going when it leaves the gun? A) answer B) v=9.8m/s h = 4.897m answer A roller coaster has a velocity of 25 m/s at the bottom of the first hill. How high was the hill? h=31.9m answer A 5 kg rock is dropped a distance of 1 m onto a spring. It compresses the spring 2 cm. What is the spring constant? k=245000N/m answer A student uses the lab apparatus shown above. A 5 kg block compresses a spring 6 cm. The spring constant is 300 N/m. What will the block's velocity be when released? answer How much work is done in stopping a 5 kg bowling ball rolling with velocity of 10 m/s? answer How much work is done compressing a spring with a 450 N/m spring constant 2 cm?