Chapter 6: Work, energy , and power 6-2 Kinetic energy 6-3 Potential energy and conservative forces 6-4 Dissipative forces 6-9 power Norah Ali Al - moneef 29/07/1438

6-1 Work In physics, work has a very specific meaning. In physics, work represents a measurable change in a system, caused by a force. Work is done on a system where an applied force results in some net displacement A force that results in no displacement does no work A displacement that results with no applied force has had no work done Norah Ali Al - moneef 29/07/1438

Work (force is parallel to distance) What do forces do? They change the speed of a mass They do work on a mass They change the energy of a mass Work (force is parallel to distance) Force (N) Work (joules) W = F x d Distance (m) Norah Ali Al - moneef 29/07/1438

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Work… W = F d cos q definition of work d - the displacement of the point of application of the force q - is the angle between the force and the displacement vectors Norah Ali Al - moneef 29/07/1438

Work Relates force to change in energy Scalar quantity Independent of time Units of Work and Energy SI unit = Joule 1 J = 1 Nm = 1 kgm2/s2 Norah Ali Al - moneef 29/07/1438

Work… W = F d cos q A force does no work on the object if the force does not move through a displacement The work done by a force on a moving object is zero when the force applied is perpendicular to the displacement of its point of application cos 90 = 0 Norah Ali Al - moneef 29/07/1438

The Scalar Product of Two Vectors Our definition of work: W = F d cos q is messy because of the cosine factor What we are actually doing is performing an operation with two vectors called the scalar product: It is also called the dot product q is the angle between A and B Norah Ali Al - moneef 29/07/1438

More About Work The sign of the work depends on the direction of the displacement relative to the force Work is positive when the projection of F onto d is in the same direction as the displacement Work is negative when the projection is in the opposite direction Work zero: W = 0 if q = 90° Work maximum if q = 0° Work minimum if q = 180° Norah Ali Al - moneef 29/07/1438

W > 0 W < 0 W = 0 Norah Ali Al - moneef 29/07/1438

Example: When Work is Zero A man carries a bucket of water horizontally at constant velocity. The force does no work on the bucket Displacement is horizontal Force is vertical cos 90° = 0 Norah Ali Al - moneef 29/07/1438

Work can be positive or negative Man does positive work lifting box Man does negative work lowering box Gravity does positive work when box lowers Gravity does negative work when box is raised Work is done on the books when they are being lifted, but no work is done on them when they are being held or carried horizontally. Norah Ali Al - moneef 29/07/1438

Work Done by a Constant Force Example: Work done on the bag by the person..  Special case: W = 0 J a) W = F d cos ( 90o ) b) Wg = m g d cos ( 90o )  Nothing to do with the motion Norah Ali Al - moneef 29/07/1438

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example Norah Ali Al - moneef 29/07/1438

Work - Example W = F d cos q The normal force ,, and the gravitational force, mg, do no work on the object cos q = cos 90° = 0 The force does do work on the object There is a component along the direction of motion W = F d cos q Norah Ali Al - moneef 29/07/1438

Example : A horizontal force F pulls a 10 kg carton across the floor at constant speed. If the coefficient of sliding friction between the carton and the floor is 0.30, how much work is done by F in moving the carton by 5m?    The carton moves with constant speed. Thus, the carton is in horizontal equilibrium.  F = f = μk N = μk mg.  Thus F = 0.3 x 10 x 9.8 = 29.4 N  Therefore work done W = FS =(29.4 Cos 0o) =147 J Norah Ali Al - moneef 29/07/1438

Norah Ali Al - moneef 29/07/1438

Example Three student push their car which run out of gas. If each student pushes with a force of 500 N and the car pushed a distance of 40 m, find the work done by the student s A man exert a horizontal force 160 N on a car that moves a distance of 10m calculate the work done by the force A student pushes on a heavy box with a force of 400 N and moves it a distance of 5 m . If the force applied at an angle of 37 shown below , find the work done by the student Norah Ali Al - moneef 29/07/1438

Work Done by a Varying Force We know that for a constant force the work is just But if the force is not constant in space we have a problem We will use an approach similar to that used to determine net displacement given a time varying velocity! Consider a force in the x-direction with a magnitude that changes depending on the exact position. We can carve that (the displacement xf-xi) displacement up into a bunch of small (very small) displacements. Norah Ali Al - moneef 29/07/1438

Work Done by a Varying Force For each of these small displacements, the force will be nearly constant, in that case, W1 » FxDx (the area of the rectangle) This will be true for all the intervals so we can form Norah Ali Al - moneef 29/07/1438

Work Done by a Varying Force, cont Again, we make the intervals vanishingly small; Therefore, The work done is equal to the area under the curve When more than one force acts on an object: or Norah Ali Al - moneef 29/07/1438

example An Eskimo returning pulls a sled as shown. The total mass of the sled is 50.0 kg, and he exerts a force of 1.20 × 102 N on the sled by pulling on the rope. How much work does he do on the sled if θ = 30° and he pulls the sled 5.0 m ? Norah Ali Al - moneef 29/07/1438

Work and Multiple Forces Suppose µk = 0.200, How much work done on the sled by friction, and the net work if θ = 30° and he pulls the sled 5.0 m ? 29/07/1438

Energy and Conservation of Energy Energy is the ability to make things change. A system that has energy has the ability to do work. Energy is measured in the same units as work because energy is transferred during the action of work. What are important kinds of energy? 1. Kinetic Energy Energy of a mass in motion Kinetic is moving energy (Dynamics) Dependant on velocity Norah Ali Al - moneef 29/07/1438

6- 2 Kinetic Energy: A force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m. The speed of the box is v1 before the push and v2 after the push. x F v1 v2 i m Norah Ali Al - moneef 29/07/1438

Kinetic energy and the Work-Kinetic energy theorem The quantity is so important that we give it a special name. The kinetic energy K of an object of mass m moving with a speed v is: Kinetic energy is the energy of a particle due to its motion. Kinetic energy is a scalar and has the same units as work (Joules) Kinetic energy and the Work-Kinetic energy theorem Norah Ali Al - moneef 29/07/1438

Kinetic Energy Energy of motion is called kinetic energy. The kinetic energy of a moving object depends on two things: mass and speed. Kinetic energy is proportional to mass. Mathematically, kinetic energy increases as the square of speed. If the speed of an object doubles, its kinetic energy increases four times. (mass is constant) Norah Ali Al - moneef 29/07/1438

An object’s kinetic energy depends on its mass and its speed. 2 The faster something is moving and the heavier it is, the more work it can do, so … If ‘v’ tripled then KE will increase by 9 times. If ‘v’ quadruples, then KE will increase by 16 times Norah Ali Al - moneef 29/07/1438

example M= 4 kg Norah Ali Al - moneef 29/07/1438

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Kinetic Energy and the Work-Kinetic Energy Theorem A change in kinetic energy is one possible result of doing work to transfer energy into a system. If the net work done on a particle is positive, the speed (and hence the kinetic energy) of the particle increases. If the net work done on a particle is negative, the speed (and hence the kinetic energy) of the object decreases. Norah Ali Al - moneef 29/07/1438

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M=6.00 kg Norah Ali Al - moneef 29/07/1438

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Example 3 : A 6.0-kg block initially at rest is pulled to the right along a horizontal, frictionless surface by a constant horizontal force of 12 N. Find the speed of the block after it has moved 3.0 m. W =F d cos 0 =12x 3x1 = 36J Δk = w 0.5m V2 =W =36 J Norah Ali Al - moneef 29/07/1438

Example What acceleration is required to stop a 1000kg car traveling 28 m/s in a distance of 100 meters? . Norah Ali Al - moneef 29/07/1438

Example A car traveling 60.0 km/h to can brake to a stop within a distance of 20.0 m. If the car is going twice as fast, 120 km/h, what is its stopping distance ? (a) (b) (1) Wnet = F d(a) cos 180o = - F d(a) = 0 – m v(a)2 / 2  - F x (20.0 m) = - m (16.7 m/s)2 / 2 (2) Wnet = F d(b) cos 180o = - F d(b) = 0 – m v(b)2 / 2  - F x (d ) = - m (33.3 m/s)2 / 2 (3) F & m are common. Thus, d = 80.0 m Norah Ali Al - moneef 29/07/1438

example Norah Ali Al - moneef 29/07/1438

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Δ k= Norah Ali Al - moneef 29/07/1438

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Potential Energy 6 – 3 potential energy and conservative forces Potential energy is associated with the position of the object within some system Potential energy is a property of the system, not the object A system is a collection of objects or particles interacting via forces or processes that are internal to the system - Potential is stored energy (Statics) Dependant on height Norah Ali Al - moneef 29/07/1438

Height object raised (m) Potential Energy Gravitational potential energy is simple to calculate Gravitational Potential Energy = weight X height Work done against gravity Mass (k g) Height object raised (m) Work (joules) W =mgh Gravity (m/sec2) Norah Ali Al - moneef 29/07/1438

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Ball speeds up Ball slows down Norah Ali Al - moneef 29/07/1438

Work Done by Gravity ball up Wg = (mg)(S) S =hf -h0 Wg =-mg(hf-h0) Yi = hf ball up Wg = (mg)(S) S =hf -h0 Wg =-mg(hf-h0) = upot,initial – upot,final mg S y Yi = h0 Yi = h x ,initial Drop ball Yf = h0 Wg = (mg)(S) S =hf -h0 Wg = mg(hf -h0) = mg(hf -h0) = Epot, final – Epot, initial y y S x x mg Norah Ali Al - moneef 29/07/1438 Yf = hf

Work Done by the Gravitational Force Work done on the ball by the gravity is: If an object is moving down, If an object is moving up, Work done by the gravity only depends on the change of height, not depends on the path. Norah Ali Al - moneef 29/07/1438

Potential Energy DU = -Dw Ugrav = +mgh DU = -Dw The change in potential energy is equal to minus the work done BY the conservative force ON the body. DU = -Dw h Work done by gravity = mg.(-h) = -mgh Therefore change in PE is DU = -Dw mg Ugrav = +mgh Lift mass m with constant velocity Norah Ali Al - moneef 29/07/1438

example What is the potential energy of a 12 kg mass raised from the ground to a to a height of 25 m? Potential Energy = weight x height change Weight = 12 x 10 = 120 N Height change = height at end - height at start = 25 - 0 = 25 m Potential energy = 120 x 25 = 3000 J   Norah Ali Al - moneef 29/07/1438

example How much potential energy is lost by a 5Kg object to kinetic energy due a decrease in height of 4.5 m PE = mgh PE = (5Kg)(10 m/s2)(4.5 m) PE = 225 Kg m2/s2 PE = 225 J Norah Ali Al - moneef 29/07/1438

example Norah Ali Al - moneef 29/07/1438

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Conservative Forces Same as direct path (-mgh) A force is conservative if the work done by it on a particle that moves between two points is the same for all paths connecting these points: otherwise the force is non-conservative. Work done by gravity w = -mgDh1+ -mgDh2+-mgDh3+… Each step height=Dh = -mg(Dh1+Dh2+Dh3 +……) = -mgh Same as direct path (-mgh) h Examples for week 2 Chapter 6 Exercises 9,12, 59 Problems 17, 18, 24, 34, 36, 37, 43, 47, 61, 64, 66, 70 Chapter 7 Exercises 12, 14, 15, 28, 29, 35 Problems 6, 34, 38, 39, 47, 49. Examples underlined are highly recommended. The large print, underlined examples are to be handed to your tutor at the tutorial the week after next. -g Norah Ali Al - moneef 29/07/1438

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Work done by gravity is independent of path taken between h0 and hf Slide block down incline h0 Wg = (mg)(S)cos S = h/cos Wg = mg(h/cos)cos Wg = mgh with h= h0-hf  h S mg hf Work done by gravity is independent of path taken between h0 and hf => The gravitational force is a conservative force. Norah Ali Al - moneef 29/07/1438

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Who is going faster at the bottom? Assume no friction Assume both have the same speed pushing off at the top same Norah Ali Al - moneef 29/07/1438

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Norah Ali Al - moneef 29/07/1438

Doing Work to Decrease Energy When you catch a ball, its kinetic energy is reduced (or absorbed) by the negative work you do on it Your muscles do negative work on your limbs and absorb energy when you land from a jump or fall Average force you must exert to absorb energy in catching a ball or landing from a jump or fall depends on how much energy must be absorbed and the displacement over which the force is absorbed Safety and protective equipment used in many sports utilizes the work/energy principle to reduce potentially damaging impact forces Examples of shock absorbing or energy absorbing materials Landing pads (gymnastics, high jumping, and pole vaulting) increase displacement of the athlete during the impact period Sand (long jumper), water (diver), midsole material in shoes (runner) Norah Ali Al - moneef 29/07/1438

Law of Conservation of Energy As energy takes different forms and changes things by doing work, nature keeps perfect track of the total. No new energy is created and no existing energy is destroyed. Norah Ali Al - moneef 29/07/1438

DU + DK = 0 Let’s check this for a body of mass m moving under gravity. Dw = DK = Kf - Ki DK = ½ mvf2 – ½ mvi2 xf vf h Dw g= -Du = -mgh +ve xi vi Dw = -DU = DK For motion under gravity you know v2 = u2 + 2as  vf2 = vi2 - 2gh mult by ½ m  ½ m vf2 = ½ mvi2 -mgh mg so DU + DK = 0 Norah Ali Al - moneef 29/07/1438

Work-Energy Theorem Work = ΔKE Gives the same answer as above. The work done on an object is equal to the change in its kinetic energy. Work = ΔKE If you know the height of the block, you can predict it’s speed at the bottom of the ramp! (doesn’t matter how LONG the ramp is) Gives the same answer as above. Norah Ali Al - moneef 29/07/1438

Weight =500 N Norah Ali Al - moneef 29/07/1438

example At B At C Norah Ali Al - moneef 29/07/1438

Example A skier slides down the frictionless slope as shown. What is the skier’s speed at the bottom? start H=40 m finish L=250 m 28.0 m/s Norah Ali Al - moneef 29/07/1438

A truck of mass 3000 kg is to be loaded onto a ship using a crane that exerts a force of 31 kN over a displacement of 2m.Find the upward speed of truck after its displacement. Norah Ali Al - moneef 29/07/1438

Norah Ali Al - moneef 29/07/1438

Work-Energy Theorem If we do work on an object and set it into motion without changing the object’s potential energy, the work done appears as kinetic energy of the object Norah Ali Al - moneef 29/07/1438

Work done on a system by external force Norah Ali Al - moneef 29/07/1438

Work – Energy Principle & Mechanical Energy Conservation If we ignore non conservative forces (friction and the such), the implication is that no non-mechanical energies are present (heat, sound, light, etc) therefore… As energy takes different forms and changes things by doing work, nature keeps perfect track of the total. No new energy is created and no existing energy is destroyed. Norah Ali Al - moneef 29/07/1438

Work and Energy If a force (other than gravity) acts on the system and does work Need Work-Energy relation or Norah Ali Al - moneef 29/07/1438

Types of Forces There are two general kinds of forces Conservative Work and energy associated with the force can be recovered Non conservative The forces are generally dissipative and work done against it cannot easily be recovered Norah Ali Al - moneef 29/07/1438

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 Examples of conservative forces include: Gravity Spring force Electromagnetic forces Norah Ali Al - moneef 29/07/1438

Non conservative Forces A force is non conservative if the work it does on an object depends on the path taken by the object between its final and starting points. Examples of non conservative forces kinetic friction Norah Ali Al - moneef 29/07/1438

Moving a block against friction at constant velocity d Work done by me = F.d Work done by friction = -f.d = -F.d Total work done = What happens if I let go? NOTHING!! Gravity is Conservative Friction is NOT!! Norah Ali Al - moneef 29/07/1438

Example Suppose the initial kinetic and potential energies of a system are 75J and 250J respectively, and that the final kinetic and potential energies of the same system are 300J and -25J respectively. How much work was done on the system by non-conservative forces? 1. 0J 2. 50J 3. -50J 4. 225J 5. -225J correct Work done by non-conservative forces equals the difference between final and initial kinetic energies plus the difference between the final and initial gravitational potential energies. W = (300-75) + ((-25) - 250) = 225 - 275 = -50J. Norah Ali Al - moneef 29/07/1438

Friction as a Non conservative Force 6-4 Dissipative forces Friction as a Non conservative Force The friction force is transformed from the kinetic energy of the object into a type of energy associated with temperature (internal energy) the objects are warmer than they were before the movement Internal Energy is the term used for the energy associated with an object’s temperature Work can be done by friction The energy lost to friction by an object goes into heating both the object and its environment Some energy may be converted into sound For now, the phrase “Work done by friction” will denote the effect of the friction processes on mechanical energy alone Norah Ali Al - moneef 29/07/1438

Friction Depends on the Path The blue path is shorter than the red path The work required is less on the blue path than on the red path Friction depends on the path and so is a non-conservative force Mechanical Energy Conservation with energy lost Norah Ali Al - moneef 29/07/1438

example a- B - Norah Ali Al - moneef 29/07/1438

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example Norah Ali Al - moneef 29/07/1438

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Another solution A ) Wgra=Fgra s cos 180 = - (mg sin 20) s =- 10 x 9.8 x5 x sin 20 = - 168 J OR Wgra= -mg ( h-h0 ) = - 10 x 9.8 x s sin 20 = - 10 x9.8 x5 x sin 20= - 168 J B ) wfriction = Ffriction s cos 180= -( mg cos 20 )s = -10 x9.8 x5 cos 20 = - 184 J C ) wapp = F s cos o = 100 x5 = 500 J wnet= wapp+ Wgra + wfriction= 500 – 168 – 184 = 148 J D ) Δ K = wnet = 148 J E ) K –K0 = 148 J Norah Ali Al - moneef 29/07/1438

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example s Θ Norah Ali Al - moneef 29/07/1438

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OR Norah Ali Al - moneef 29/07/1438

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Norah Ali Al - moneef 29/07/1438

example You are towing a car up a hill with constant velocity. A )The work done on the car by the normal force is: 1. positive 2. negative 3. zero W T FN V correct The normal force is perpendicular to the displacement, hence, does no work. b )The work done on the car by the gravitational force is: 1. positive 2. negative 3. zero With the surface defined as the x-axis, the x component of gravity is in the opposite direction of the displacement, therefore work is negative. correct C ) The work done on the car by the tension force is: 1. positive 2. negative 3. zero correct Tension is in the same direction as the displacement. Norah Ali Al - moneef 29/07/1438

6-9 power Norah Ali Al - moneef 29/07/1438

Power Takes more power to run up the stairs than to walk up the stairs, but the energy consumed is the same in either case Unit is the WATT A Watt is a newton--meter per second Bigger units are kilowatts and megawatts Utility sells energy in kilowatt-hours 1 KWh = 1000 Joules/second times 3600 Seconds = 3.6 X 106 Joule Norah Ali Al - moneef 29/07/1438

Norah Ali Al - moneef 29/07/1438

Example: Killer whales are able to accelerate up to 30 mi/h in a matter of seconds. Disregarding the considerable drag force of water, calculate the average power a killer whale with mass 8000 kg would need to generate to reach a speed of 12.0 m/s in 6.00 s? Norah Ali Al - moneef 29/07/1438

example A 100-watt light bulb that is on 10-hours a day for 30 days. How much energy will it use? How much did it cost to operate that lamp if the price is $.09053 per kWh? · 10 hours per day x 10 days = 300 hours (energy is all about time of operation) · 300 Hours x 100 Watts = 30,000 watt hours (energy is also about connected load or Watts) · A kilowatt (kW) is 1000 watts · 30,000 watt hours ÷ 1000 watts = 30 kilowatt hours (kWh as on the electric bill 30 kWh x $.09053 per kWh = $2.72 cost of operation Norah Ali Al - moneef 29/07/1438

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? Since the speed is constant, so a = 0 where M = total mass of the system M= 1000+800=1800 Kg Norah Ali Al - moneef 29/07/1438

example OR Norah Ali Al - moneef 29/07/1438

example A 5 Kg Cart is pushed by a 30 N force against friction for a distance of 10m in 5 seconds. Determine the Power needed to move the cart. Norah Ali Al - moneef 29/07/1438

example In a rifle barrel, a 15.0 g bullet is accelerated from rest to a speed of 780 m/s. a)Find the work that is done on the bullet.? Using Work-Kinetic Energy Theorem, W = ΔKE W = KEf - KEi W = 1/2(0.015 kg)(780 m/s)2 - 0 W = 4.56 kJ b) If the rifle barrel is 72.0 cm long, find the magnitude of the average total force that acted on the bullet. Using W = Fcosθd, F = (4.56 kJ) / (0.72 m) F = 6.33 kN Norah Ali Al - moneef 29/07/1438

Example: The minimum work required to raise a 800 N person up 10 m, is: W = F d W = (800 N) (10 m) = 8000 J If this work is done in 60 sec, then what is the power? Norah Ali Al - moneef 29/07/1438

example A car of mass 500 kg is traveling along a horizontal road .the engine of the car is working at a constant rate of 5kw .the total Resistance to motion is constant and is 250 N . What is the acceleration of the car when its speed is 5 m/s Fk= m/s Norah Ali Al - moneef 29/07/1438

example Norah Ali Al - moneef 29/07/1438

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Summary of Chapter 6 Work: Kinetic energy is energy of motion: Work-energy principle: The net work done on an object equals the change in its kinetic energy. Conservative force: work depends only on end points Gravitational potential energy: Ugrav = mg (h –h0 ). Total mechanical energy is the sum of kinetic and potential energies. Additional types of energy are involved when nonconservative forces act. Total energy (including all forms) is conserved. Power: rate at which work is done, or energy is transformed: Δ k = - Δ u K+u =ko +uo K+u =ko +uo+ wa or Norah Ali Al - moneef 29/07/1438