Work and Energy Work: The word looks the same, it spells the same but has different meaning in physics from the way it is normally used in the everyday language

Definition of work W done by a constant force F exerted on an object through distance d θ d F θ d d Only the component that acts in the same direction as the motion is doing work on the box. Vertical component is just trying to lift the object up. work = force along distance × the distance moved work = force × distance moved × cos of the angle between them

Units The SI unit for work is the newton–metre and is called the joule named after the 19th Century physicist James Prescott Joule. 1 J (Joule) = 1N x 1 m Work is a scalar (add like ordinary numbers)

A force is applied. Question: Is the work done by that force? Work - like studying very hard, trying to lift up the car and get completely exhausted, holding weights above head for half an hour is no work worth mentioning in physics. According to the physics definition, you are NOT doing work if you are just holding the weight above your head (no distance moved) you are doing work only while you are lifting the weight above your head (force in the direction of distance moved)

Who’s doing the work around here? NO WORK WORK If I carry a box across the room I do not do work on it because the force is not in the direction of the motion (cos 900 = 0)

Work done by a force F is zero if: θ = 00 00< θ <900 θ = 900 900< θ <1800 θ = 1800 cos θ = 1 cos θ = + cos θ = 0 cos θ = – cos θ = –1 Work done by a force F is zero if: force is exerted but no motion is involved: no distance moved, no work force is perpendicular to the direction of motion (cos 900 = 0) d F F d F motion normal force tension in the string gravitational force

Work done by force F is: positive W = Fd negative when the force and direction of motion are generally in the same directions (the force helps the motion – can increase kinetic energy by increasing speed) 00< θ < 900 → cos θ = + cos 00 = 1 W = Fd negative when the force and direction of motion are generally in the opposite directions (force opposes the motion – decreases kinetic energy by decreasing speed) 900< θ < 1800 → cos θ = – cos 1800 = –1 W = - Fd (the work done by friction force is always negative)

Mike is cutting the grass using a human-powered lawn mower Mike is cutting the grass using a human-powered lawn mower. He pushes the mower with a force of 45 N directed at an angle of 41° below the horizontal direction. Calculate the work that Mike does on the mower in pushing it 9.1 m across the yard. EXAMPLE F = 45 N d = 9.1 m θ = 410 d 410 F W = Fd cos θ = 310 J

Forward force is 200 N. Friction force is 200 N. The distance moved is 200 km. Find the work done by forward force F on the car. the work done by friction force Ffr on the car. the net work done on the car. EXAMPLE F = 200 N Ffr = 200 N d = 2x105 m WF = Fd cos 00 = 4x107 J Wfr = Ffr d cos 1800 = - 4x107 J the net work done on the car means the work done by net force on the car. It can be found as: W = WF + Wfr = 0 or W = Fnet d cos θ = 0 (Fnet = 0)

Work done by a varying force - graphically W = Fd cos θ applies only when the force is constant. Force can vary in magnitude or direction during the action. Examples: 1) rocket moving away from the Earth – force of gravity decreases 2.) varying force of the golf club on a golf ball Work done is determined graphically.

The area under a Force - distance graph equals The lady from the first slide is pulling the car for 2 m with force of 160 N at the angle of 60o , then she gets tired and lowers her arms behind her at an angle of 45o pulling it now with 170 N for next 2 m. Finally seeing the end of the journey she pulls it horizontally with the force of 40 N for 1 m. Work done by her on the car is: W = (160 N)(cos 60o)(2m) +(170 N)(cos 45o)(2m) + (40 N)(cos 0o)(1m) W = 80x2 + 120x2 + 40x1 = pink area + green area + blue area = 440 J http://www.kcvs.ca/map/java/applets/workEnergy/applethelp/lesson/lesson.html#1 In general: The area under a Force - distance graph equals the work done by that force

Gravitational potential energy, PE = mg h Imagine a mass m lifted a vertical distance h following two different paths. straight up: – minimum force needed is F1 = mg. – the work done against gravitational force is: W = F1 h cos 00 W = mg h along a ramp a distance d (no friction) – minimum force needed is F2 = mg sinθ. W = F2 d cos 00 = mg sinθ d F2 h d F1 θ mg the work W = mg h is stored as potential energy of the object. PE = mg h It depends on VERTICAL distance from initial position not on the path taken.

A ramp can reduce the force – can make life easier θ 1. W = F× h = mg × h 2. W = F × d = mg sinθ × h/sinθ = mg × h W = F× h or F × d (along the ramp)

Kinetic energy, KE = ½ m v2 A moving object possesses the capacity to do work. A hammer by virtue of its motion can be used to do work in driving a nail into a piece of wood. How much work a moving object is capable of doing? Let’s find out. Imagine an object of mass m accelerated from rest by a resultant force, F. After traveling a distance d the mass has velocity v. The work done is: W = Fd cos 00 = Fd = mad From v2 = u2 + 2ad → ad = v2/2 and W = ½ mv2 The work done (Fd) by the force F in moving the object a distance d is now expressed in terms of the properties of the body and its motion. The quantity ½ mv2 is called the kinetic energy (KE) of the body and is the energy that a body possesses by virtue of its motion. The kinetic energy of a body essentially tells us how much work the body is capable of doing.

Useful relationship between momentum and kinetic energy: KE depends on the reference frame in which it is measured. When you are sleeping, you have zero KE realtive to your bed. But relative to the sun, you have KE = ½ (60 kg) (30000 m/s)2

Summary: Work done on an object by applied force against gravitational force (when the net force is zero, so there is no acceleration) is stored as gravitational potential energy. PE depends on the reference frame in which it is measured, so we keep in mind some reference level, like desk, lower reservoir of water, ground,…. PE = mg ∆h Work done on a spring by applied force against spring force (when the net force is zero, so there is no acceleration) is stored as elastic potential energy… EPE = ½ kx2 If the net force acting on a body is not zero, then: The work done by net force on a body is equal (results) to the change in the kinetic energy of the body: W = ∆KE = KEf – KEi

Units W = Fd (W) = (1N)(1m) = 1 kg m2 s-2 = 1 J PE = mgh (PE) = (1kg)(1 m s-2 )(1m) = 1 kg m2 s-2 = 1 J EPE = ½ kx2 (EPE) = (1N/1m)(1m2) = 1 kg m2 s-2 = 1 J KE = ½ mv2 (KE) = (1kg)(1m/s)2 = 1 kg m2 s-2 = 1 J Even in units we see that the work and energy are equivalent.

W = ∆ KE = ½ m v2 – 0 = ½ (30 kg)(2)2 = 60 J A father pushes his child on a sled on level ice, a distance 5 m from rest, giving a final speed of 2 m/s. If the mass of the child and sled is 30 kg, how much work did he do? W = ∆ KE = ½ m v2 – 0 = ½ (30 kg)(2)2 = 60 J other way: W = Fd cosθ = Fd F = ma F = 12 N W = 60 J What is the average force he exerted on the child? W = Fd = 60 J, and d = 5 m, so F = 60/5 = 12 N EXAMPLE

How much work must this friction force do in order to stop the car? A 1000-kg car going at 45 km/h. When the driver slams on the brakes, the road does work on the car through a backward-directed friction force. How much work must this friction force do in order to stop the car? W = ∆ KE = 0 – ½ m u2 = – ½ (1000 kg) (45 x1000 m/3600 s)2 W = – 78125 J = – 78 kJ (the – sign just means the work leads to a decrease in KE) EXAMPLE d u Ffr (work done by friction force) v = 0

W = – 78125 J The net force acting on the car is F F = 260 x 103 N Instead of slamming on the brakes the work required to stop the car is provided by a tree!!! What average force is required to stop a 1000-kg car going at 45 km/h if the car collapses one foot (0.3 m) upon impact? EXAMPLE W = – 78125 J W = – F d = – ½ m u2 The net force acting on the car is F F = 260 x 103 N corresponds to 29 tons hitting you OUCH Do you see why the cars should not be rigid. Smaller collapse distance, gretaer force, greater acceleration. For half the distance force would double!!!! OUCH, OUCH a = (v2 - u2 )/2d = - 520 m/s2 HUGE!!!!

The principle of energy conservation !!!!!! we shall see in nuclear physics that this should actually be the principle of mass-energy conservation To demonstrate the so-called principle of energy conservation we will solve a dynamics problem in two different ways, one using Newton’s laws and the kinematics equations and the other using the principle of energy conservation. An object of mass 4.0 kg slides from rest without friction down an inclined plane. The plane makes an angle of 30° with the horizontal and the object starts from a vertical height of 0.5 m. Determine the speed of the object when it reaches the bottom of the plane.

point B: h = 0 , object has gained KE mg 300 B A h=0.5m v=? u=0 point A: h = 0.5 m, KE = 0 (u = 0) point B: h = 0 , object has gained KE d Method 2: Energy conservation Method 1: Newton’s law As the object slides down the plane its PE becomes transformed into KE. If we assume that no energy is ‘lost’ we can write v2 = u2 + 2ad = 2gsinθ PEA = KEB mgh = ½ mv2 v = 3.2 m/s v = 3.2 m/s Note that the mass of the object does not come into the question, nor does the distance travelled down the plane, only vertical height.

The second solution involves making the assumption that potential energy is transformed into kinetic energy and that no energy is converted into other forms of energy (heat, sound,…). This is ‘Law of conservation of energy’, this means the energy is conserved. Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. When using the energy principle we are only concerned with the initial and final conditions and not with what happens in between. No time included in energy principle. Very powerful tool of enormous importance. Clearly in this example it is much quicker to use the energy principle. This is often the case with many problems and in fact with some problems the solution can only be achieved using energy considerations.

If it is transformed into thermal energy, we say it is dissipated If it is transformed into thermal energy, we say it is dissipated. The object can’t keep that energy. It is shared with environment. and now problems and examples. Some of them would be very, very hard to solve using kinematic equations and Newton’s laws.

Dropping down from a pole. As he dives, PE becomes KE. Total energy is always constant, equal to initial energy. If accounted for air resistance, then how would the numbers change? EXAMPLE In presence of air, some energy gets transformed to heat (which is random motion of the air molecules). Total energy at any height would be PE + KE + heat, so at a given height, the KE would be less than in vacuum. What happens when he hits the ground? Just before he hits ground, he has large KE (large speed). This gets transformed into heat energy of his hands and the ground on impact, sound energy, and energy associated with deformation .

Amusement park physics the roller coaster is an excellent example of the conversion of energy from one form into another work must first be done in lifting the cars to the top of the first hill. the work is stored as gravitational potential energy you are then on your way! EXAMPLE PE is being transformed into KE and vice versa

Up and down the track PE PE PE + KE EXAMPLE PE PE PE + KE If friction is negligible the ball will get up to the same height on the right side.

Three balls are thrown from the top of the cliff along paths A, B, and C with the same initial speed (air resistance is negligible). Which ball strikes the ground below with the greatest speed? a. A b. B c. C d. All strike with the same speed EXAMPLE h The initial PE + KE of each ball is: mgh + ½ mu2. This amount of energy becomes KE before impact ½ mv2. mgh + ½ mu2 = ½ mv2 m cancels The speed of impact for each ball is the same: It depends ONLY on HEIGHT and initial SPEED, not mass, not path !!!!!

A child of mass m is released from rest at the top of a water slide, at height h = 8.5 m above the bottom of the slide. Assuming that the slide is frictionless because of the water on it, find the child’s speed at the bottom of the slide. Do you think you could use kinematics equations and Newton’s laws? Try it. Good luck ! EXAMPLE mgh = ½ mv2 m cancels on both sides a baby, an elephant and you would reach the bottom at the same speed !!!!! And, by the way, it is the same speed as you were to fall straight down from the same height. v2 = u2 + 2gh = 2gh

EXAMPLE Work is a way of transferring energy from one form to another, but itself is not a form of energy. All types of energies and work have the same units. Everything can be transformed into each other. When work is done on an object, energy is transferred to that object. Example: A spring at the bottom of a slide is compressed by an external force. A parent releases the spring when a child sits on it. EPE is transferred to the child by spring force doing the work which is transformed into KE of the child. On the way up, KE is being transformed into PE. This energy is what enables that child to then do work. How? First that PE has to be transformed back into KE – child slides down a slide – KE of the child can do work on the waiting parent – it can knock him over. Parent does the work on the ground.

A car (toy car – no engine) is at the top of a hill on a frictionless track. What must the car’s speed be at the top of the first hill if it can just make it to the top of the second hill? EXAMPLE v2 = 0 PE1 + KE1 = PE2 + KE2 v1 mgh1 + ½ mv12 = mgh2 + ½ mv22 m cancels out ; v2 = 0 gh1 + ½ v12 = gh2 40m 20m v12 = 2g( h2 – h1) v1 = 22 m/s

as shown. Its speed at the lowest point B is: A simple pendulum consists of a 2.0 kg mass attached to a string. It is released from rest at A as shown. Its speed at the lowest point B is: 1.85 m A B EXAMPLE PEA + KEA = PEB + KEB  

● Conservation of energy law AGAIN – WITH FRICTION INCLUDED All previous examples were neglecting friction force, air resistance … What happens if the friction force acts? Can we still use conservation of energy principle? YES, WE CAN

on the object as it slides down. ● Example: An object of mass 4.0 kg slides 1.0 m down an inclined plane starting from rest. Determine the speed of the object when it reaches the bottom of the plane if a. friction is neglected b. constant friction force of 16 N acts on the object as it slides down. all PE was transformed into KE Friction converts part of KE of the object into heat energy. This energy equals to the work done by the friction. We say that the frictional force has dissipated energy. initial energy = final energy initial energy – Ffr d = final energy Wfr = – Ffr d decreases KE of the object PEA = KEB PEA – Ffr d = KEB mgh = ½ mv2 mgh – Ffr d = ½ mv2 20 – 16 = 2.5 v2 v = 1.4 m/s

Types of energy and energy transformations We usually classify energy as the following forms: 1 Kinetic energy 2 Gravitational potential energy 3 Elastic potential energy 4 Thermal or heat energy 5 Light energy 6 Sound energy 7 Chemical energy 8 Electrical energy 9 Magnetic energy 10 Nuclear energy There are energy changes happening all around us all of the time. Here are a few examples to think about.

In any given transformation, some of the energy is almost inevitably changed to heat. • A bouncing ball The gravitational potential energy is changed to kinetic energy and back again with each bounce losing energy due to work done by air friction and nonelastic contact with the ground . Eventually the ball comes to rest and all the energy has become low grade heat. • An aeroplane taking off As fuel is burned in the engines, chemical energy is converted to heat, light and sound. The plane accelerates down the runway and its kinetic energy increases. There will be heat energy due to friction between the tyres and the runway. As the plane takes off and climbs into the sky, its gravitational potential energy increases.

• A nuclear power station Nuclear energy from the uranium fuel changes to thermal energy that is used to boil water. The kinetic energy of the steam molecules drives turbines, and as the kinetic energy of the turbines increases, it interacts with magnetic energy to give electrical energy. • A laptop computer The electrical energy form the mains or chemtcal energy from the battery is changed to light on the screen and sound through the speakers. There is kinetic energy in the fan, magnetic energy in the motors, and plenty of heat is generated. • A human body Most of the chemical energy from our food is changed to heat to keep us alive. Some is changed to kinetic energy as we move, or gravitational potential energy if we climb stairs. There is also elastic potential energy in our muscles, sound energy when we talk and electrical energy in our nerves and brain.

End of Day 1

Power ◘ Power is the work done in unit time or energy converted in unit time measures how fast work is done or how quickly energy is converted. Power is a scalar quantity. ◘ Units: A 100 W light bulb converts electrical energy to heat and light at the rate of 100 J every second. Sometimes you’ll see power given in kW or even MW.

There is another way to calculate power Calculate the power of a worker in a supermarket who stacks shelves 1.5 m high with cartons of orange juice, each of mass 6.0 kg, at the rate of 30 cartons per minute. P = 45 W There is another way to calculate power Power is equal to force times velocity, providing that both force and velocity are constant and in the same direction. Constant velocity with a force applied ?????? That’s because we are interested in one force only. Net force is obviously zero. Like power of the engine of the car.

Efficiency ◘ Efficiency is the ratio of how much work, energy or power we get out of a system compared to how much is put in. ◘ No units ◘ Efficiency can be expressed as percentage by multiplying by 100%. ◘ No real machine or sysem can ever be 100% efficient, because there will always be some energy changed to heat due to friction, air resistance or other causes.

How much energy is converted into heat per second. A car engine has an efficiency of 20 % and produces an average of 25 kJ of useful work per second. How much energy is converted into heat per second. Ein = 125000 J heat = 125 kJ – 25 kJ = 100 kJ

Spring/Tension force – Hooke’s law Holding one end and pulling the other produces a tension (spring) force in the spring. You’ll notice as you pull the spring, that the further you extend the spring, then the greater the force that you have to exert in order to extend it even further. As long as the spring is not streched beyond a certain extension, called elastic limit, the force is directly proportional to the extension. Beyond this point the proportionality is lost. If you stretch it more, the spring can become permanently deformed in such a way that when you stop pulling, the spring will not go back to its original length.

F = kx Compressed Unstretched Stretched spring L Stretched x pulling force F Compressed pushing In the region of proportionality we can write F = kx k is a constant whose value depends on the particular spring. For this reason k is called the spring constant. It measures stiffness of the spring in Newtons per metre.

Example: k = 20 N/m L = 10 cm How much force do you have to exert if you want to extend the spring for 1cm 2 cm 3 cm EXAMPLE F = kx = (20 N/m)(0.01 m) = 0.2 N F = kx = (20 N/m)(0.02 m) = 0.4 N F = kx = (20 N/m)(0.03 m) = 0.6 N proportionality Note: The spring force has the same value, but it is in the opposite direction !!!!!

F is the spring force exerted by the spring Spring force – Hooke’s law: F = – kx The spring force is the force exerted by a compressed or stretched spring upon any object that is attached to it. Note: The spring force has the same value, but it is in the opposite direction !!!!! x is the displacement (extension/compression) of the spring's end from its equilibrium position F is the spring force exerted by the spring k is a constant called spring constant (in SI units: N/m).  

The work done by a non-constant applied force on a Hooke’s spring is found from the area under the graph F vs. x Work done by a force F = kx when extending a spring from extension x1 to x2 is: extension x2 x1 F kx2 kx1 – Work done by a force F = kx when extending a spring from extension 0 to x is: W = ½ kx2

Energy, E In physics energy and work are very closed linked; in some senses they are the same thing. If an object has energy it can do a work. On the other hand the work done on an object is converted into energy. work done = change in energy W = ∆ E

Elastic potential energy, EPE If some force stretches a spring by extension x, the work done by that force is ½ kx2. Since work is the transfer of energy, we say that the energy was transferred into the spring, and that work is now stored in stretched spring as elastic potential energy. EPE = ½ kx2 A spring can be stretched or compressed. The same mathematics holds for stretching as for compressing springs. Just imagine how much energy is stored in the springs of this scale.