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• Lesson 1: Basic Terminology and Concepts • Work – Definition and Mathematics of Work – Calculating the Amount of Work Done by Forces • • • • Potential Energy Kinetic Energy Mechanical Energy Power Definition and Mathematics of Work • In physics, work is defined as a force _________ acting upon an object to cause displacement ____________ a __________________. Work is being done Work Workisisnot notbeing beingdone done Let’s practice – work or no work 1. A student applies a force to a wall and becomes exhausted. 2. A calculator falls off a table and free falls to the ground. 3. A waiter carries a tray full of beverages above his head by one arm across the room 4. A rocket accelerates through space. Calculating the Amount of Work Done by Forces W = F∙d∙cosθ F θ • F - is the force in Newton, which causes the displacement of the object. • d - is the displacement in meters • θ = angle between force and displacement • W - is work in N∙m or Joule (J). 1 J = 1 N∙m = 1 kg∙m2/s2 d Fy θ d F scalar • Work is a _____________ quantity Fx • Work is independent of time the force acts on the object. Only the horizontal component of the force (Fcosθ) causes a horizontal displacement. W = F∙d∙cosθ positive, negative or zero work Positive work negative work - force acts in the direction opposite the objects motion in order to slow it down. no work To Do Work, Forces Must Cause Displacements W = F∙d∙cosθ = 0 The angle in work equation W = F∙d∙cosθ • The angle in the equation is the angle between the force and the displacement vectors. F & d are in the same direction, θ is 0o. d F example • A 20.0 N force is used to push a 2.00 kg cart a distance of 5.00 meters. Determine the amount of work done on the cart by the force. 20.0 N example • How much work is done in lifting a 5.0 kg box from the floor to a height of 1.2 m above the floor? example • A 2.3 kg block rests on a horizontal surface. A constant force of 5.0 N is applied to the block at an angle of 30.o to the horizontal; determine the work done on the block a distance of 2.0 meters along the surface. 5.0 N 30o 2.3 kg practice • Matt pulls block along a horizontal surface at constant velocity. The diagram show the components of the force exerted on the block by Matt. Determine how much work is done against friction. 6.0 N F 8.0 N 3.0 m example • A neighbor pushes a lawnmower four times as far as you do but exert only half the force, which one of you does more work and by how much? Force vs. displacement graph Example: a block is pulled along a table with 10. N over a distance of 1.0 m. W = Fd = (10. N)(1.0 m) = 10. J height base area Force (N) • The area under a force versus displacement graph is the work done by the force. work Displacement (m) Potential energy • An object can store energy as the result of Potential energy its position. ________________________ is the stored energy of position possessed by an object. • Two form: – Gravitational – Elastic Gravitational potential energy • Gravitational potential energy is the energy stored in an object as the result of its _________________________ vertical position (height). gravitational • The energy is stored as the result of the _____________ attraction of the Earth for the object. • The work done in raising an object must result in an gravitational potential energy increase in the object's _______________________ • The gravitational potential energy of an object is dependent on three variables: – The mass of the object – The height of the object – The gravitational field strength PEgrav = m∙g∙h • Equation: ______________________ – m: mass, in kilograms – h: height, in meters – g: acceleration of gravity = 9.81 m/s2 GPE • GPE = mgh • The equation shows that . . . • the more mass a body has • or the stronger the gravitational field it’s in • or the higher up it is • . . . the more gravitational potential energy it’s got. GPE and work done by gravity • When an object falls, gravity does positive work. Object loses GPE. hi • Wgrav = mg(hi – hf) • Wgrav = - mg(hf – hi) = - mg∆h As long as the falling height is the same, gravity did The same amount of work regardless of which path is taken. hf GPE and work against gravity • When an object is raised against gravity at constant speed (no change in kinetic energy), gravity does negative work. Object gains GPE. hf • Work done against gravity = mg∆h As long as the object is raised to the same height, work done against gravity is the same regardless of which path is taken. hi The increase in an object's potential energy equals the work done in raising an object Each path up to the seat top requires the same amount of work. The amount of work done by a force on any object is given by the equation W = F∙d∙cosθ where F is the force, d is the displacement and θ is the angle between the force and the displacement vector. In all three cases, θ equals to 0o example • The diagram shows points A, B, and C at or near Earth’s surface. As a mass is moved from A to B, 100. joules of work are done against gravity. What is the amount of work done against gravity as an identical mass is moved from A to C? Unit of energy • The unit of energy is the same as work: Joules _______ • 1 joule = 1 (kg)∙(m/s2)∙(m) = 1 Newton ∙ meter • 1 joule = 1 (kg)∙(m2/s2) Work and energy has the same unit Gravitational potential energy is relative • To determine the gravitational potential energy of an object, a _______ zero height position must first be assigned. ground • Typically, the ___________ is considered to be a position of zero height. • But, it doesn’t have to be: – It could be relative to the height above the lab table. – It could be relative to the bottom of a mountain – It could be the lowest position on a roller coaster example • How much potential energy is gained by an object with a mass of 2.00 kg that is lifted from the floor to the top of 0.92 m high table? • The graph of gravitational potential energy vs. vertical height for an object near Earth's surface gives the weight of the object. The weight of the object is the slope of the line. Weight = __________ Elastic potential energy • Elastic potential energy is the energy elastic stored in ______________ materials as the result of their stretching or compressing. • Elastic potential energy can be stored in – Rubber bands – Bungee cores – Springs – trampolines Hooke’s Law F = kx Spring force = spring constant x displacement • F in the force needed to displace (by stretching or compressing) a spring x meters from the equilibrium (relaxed) position. The SI unit of F is Newton. • k is spring constant. It is a measure of stiffness of the spring. The greater value of k means a stiffer spring because more force is needed to stretch or compress it that spring. The Si units of k are N/m. depends on the material made up of the spring. k is in N/m • x the distance difference between the length of stretched/compressed spring and its relaxed (equilibrium) spring. example • A spring has a spring constant of 25 N/m. What is the minimum force required to stretch the spring 0.25 meter from its equilibrium position? example • The graph below shows elongation as a function of the applied force for two springs, A and B. Compared to the spring constant for spring A, the spring constant for spring B is 1. smaller 2. larger 3. the same Elastic potential energy in a spring • Elastic potential energy is the Work done on the spring. PEs = Favg∙d = Favg∙x = (½ k∙x)∙x = ½ kx2 Note: F is the average force PEs = ½ k∙x2 – k: spring constant – x: amount of compression or extension relative to equilibrium position Elastic potential energy Elastic potential energy is directly proportional to x2 elongation example • A spring has a spring constant of 120 N/m. How much potential energy is stored in the spring as it is stretched 0.20 meter? example • The unstretched spring in the diagram has a length of 0.40 meter and a spring constant k. A weight is hung from the spring, causing it to stretch to a length of 0.60 meter. In terms of k, how many joules of elastic potential energy are stored in this stretched spring? example • Determine the potential energy stored in the spring with a spring constant of 25.0 N/m when a force of 2.50 N is applied to it. example • As shown in the diagram, a 0.50-meter-long spring is stretched from its equilibrium position to a length of 1.00 meter by a weight. If 15 joules of energy are stored in the stretched spring, what is the value of the spring constant? example • A 10.-newton force is required to hold a stretched spring 0.20 meter from its rest position. What is the potential energy stored in the stretched spring? • A force of 0.2 N is needed to compress a spring a distance of 0.02 meter. What is the potential energy stored in this compressed spring? Kinetic energy motion • Kinetic energy is the energy of _______. • An object which has motion - whether it be vertical or horizontal motion - has kinetic energy. • The equation for kinetic energy is: 2 KE = ½ mv __________________ – Where KE is kinetic energy, in joules – v is the speed of the object, in m/s – m is the mass of the object, in kg Kinetic Energy • KE = ½ m v 2 • The equation shows that . . . • the more mass a body has • or the faster it’s moving • . . . the more kinetic energy it has. • KE is directly proportional to m, so doubling the mass doubles kinetic energy, and tripling the mass makes it three times greater. Kinetic energy Kinetic energy • KE is proportional to v 2, so doubling the speed quadruples kinetic energy, and tripling the speed makes it nine times greater. speed mass Example • A 55 kg toy sailboat is cruising at 3 m/s. What is its kinetic energy? Note: Kinetic energy (along with every other type of energy) is a scalar, not a vector! example • An object moving at a constant speed of 25 meters per second possesses 450 joules of kinetic energy. What is the object's mass? example • a. b. c. d. A cart of mass m traveling at a speed v has kinetic energy KE. If the mass of the cart is doubled and its speed is halved, the kinetic energy of the cart will be half as great twice as great one-fourth as great four times as great example • Which graph best represents the relationship between the kinetic energy, KE, and the velocity of an object accelerating in a straight line? a b c d Mechanical Energy • Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position) or both. The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy TME = KE + PEg + PEs Mechanical Energy as the Ability to Do Work • Any object that possesses mechanical energy - whether it is in the form of potential energy or kinetic energy - is able to do work. • The diagram shows the motion of Brie as she glides down the hill and makes one of her record-setting jumps. TME = TME = TME = TME = TME = Power • Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation. • The standard metric unit of power is the Watt. • All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other objects. • The power rating of a car relates to how rapidly the car can be accelerated. • Some people are more power-full than others. That is, some people are capable of doing the same amount of work in less time or more work in the same amount of time example • Ben Pumpiniron elevates his 80-kg body up the 2.0-meter stairwell in 1.8 seconds. What is his power? It can be assumed that Ben must apply an (80 kg x 9.81 m/s2) -Newton downward force upon the stairs to elevate his body. Another equation for power example • Two physics students, Will N. Andable and Ben Pumpiniron, are in the weightlifting room. Will lifts the 100-pound barbell over his head 10 times in one minute; Ben lifts the 100-pound barbell over his head 10 times in 10 seconds. Which student does the most work? ______________ Which student delivers the most power? ______________ Explain your answers. example • When doing a chin-up, a physics student lifts her 42.0-kg body a distance of 0.25 meters in 2 seconds. What is the power delivered by the student's biceps? kilowatt-hour is unit for energy • Your household's monthly electric bill is often expressed in kilowatt-hours. One kilowatt-hour is the amount of energy delivered by the flow of l kilowatt of electricity for one hour. Use conversion factors to show how many joules of energy you get when you buy 1 kilowatt-hour of electricity. • Lesson 2: The Work-Energy Theorem • Internal vs. External Forces • The Work-Energy Connection – Analysis of Situations Involving External Forces – Analysis of Situations in Which Mechanical Energy is Conserved – Application and Practice Questions • Bar Chart Illustrations What is energy? ability • Energy is the __________to do work. scalar • Energy is a _________quantity. • When work is done on or by a system, the total energy of the system is changed. Work-Energy theorem • The net work done (work done by net force) on an object equals to the change in the object’s kinetic energy Wnet = ∆KE = KE2 – KE1 Fnetd = KE2 – KE1 Practice Problem #1 • A 1000-kg car traveling with a speed of 25 m/s skids to a stop. The car experiences an 8000 N force of friction. Determine the stopping distance of the car. Practice Problem #2 • At the end of the Shock Wave roller coaster ride, the 6000-kg train of cars (includes passengers) is slowed from a speed of 20 m/s to a speed of 5 m/s over a distance of 20 meters. Determine the braking force required to slow the train of cars by this amount. • The above problems have one thing in common: there is a force which does work over a distance in order to remove mechanical energy from an object. • The force acts opposite the object's motion and thus does negative work which results in a loss of the object's total amount of mechanical energy. In each situation, the work is related to the kinetic energy change. TMEi + Wext = TMEf KEi + Wext = 0 J ½ •m•vi2 + F•d•cos(180o) = 0 J F•d = ½ •m•vi2 d ~ v i2 Stopping distance is dependent upon the square of the velocity. Stopping distance and initial velocity Wnet = ∆KE Wnet = 0 - ½ •m•vi2 Ff•d = - ½ •m•vi2 • Ff = μFnorm = μmg • - μmg•d = - ½ mvi2 • d = vi2 / 2μg d ~ v i2 practice (m/s) Stopping Distance (m) 0 m/s 0 5 m/s 4m 10 m/s 15 m/s 20 m/s 25 m/s Conservative vs. non-conservative Forces • There are a variety of ways to categorize all the types of forces. 1. Contact force: Forces that arise from the physical contact of two objects. 2. Field force exist between objects, even in the absence of physical contact between the objects. • We can also categorize forces based upon whether or not their presence is capable of changing an object's total mechanical energy. 1. Conservative force can never change the total mechanical energy of an object 2. Non-conservative forces will change the total mechanical energy of the object • The conservative forces include the gravity forces, spring force, magnetic force, electrical force. • We will simply say all the other forces are nonconservative forces, such as applied force, normal force, tension force, friction force, and air resistance force. conservative forces Non-conservative forces Fgrav Fspring Fapp Ffrict Ften FNorm Work energy theorem Wnet = ∆KE Wgrav + Wspring + Wother = ∆KE Wgrav = work done by Gravity Wgrav = mg(hi – hf) Wspring = work done by spring Wspring = ½ k(xi2 – f mghi – mghf + ½ kxi2 – ½kxf2 + Wother = ½ kvf2 – ½kvi2 mghi + ½ kxi2 + ½kvi2 + Wother = ½ kvf2 + mghf + ½kxf2 PEgi + PEsi + KEi + Wother = PEgf + PEsf + KEf PEgi + PEsi + KEi + Wother = PEgf + PEsf + KEf TMEi + Wother = TMEf Wother = TMEf - TMEi • When net work is done upon an object by an non-conservative force, the total mechanical energy (KE + PE) of that object is changed. – If the work is positive work, then the object will gain energy. – If the work is negative work, then the object will lose energy. – The gain or loss in energy can be in the form of potential energy, kinetic energy, or both. – The work done will be equal to the change in mechanical energy of the object. TMEi + Wother = TMEf Wother = TMEf - TMEi • When the only type of force doing net work upon an object is conservative force (Wother = 0), the total mechanical energy (KE + PE) of that object remains constant. TMEf = TMEi. In such cases, the object's energy changes form. • For example, as an object is "forced" from a high elevation to a lower elevation by gravity, some of the potential energy of that object is transformed into kinetic energy. Yet, the sum of the kinetic and potential energies remain constant. • When Wother = 0, TMEi = TMEf, energy is transformed or changes its form from kinetic energy to potential energy (or vice versa); the total amount present is conserved i.e., always the same. In Summary • When only conservative forces do work – TME is conserved, – TME1 = TME2 • When non-conservative forces do work – TME changes, – Wother = TME2 – TME1 example example example example example • A block weighing 15 N is pulled to the top of an incline that is 0.20 meter above the ground, as shown below. If 5.0 joules of work are needed to pull the block the full length of the incline, how much work is done against friction? example • A shopping cart full of groceries is sitting at the top of a 2.0-m hill. The cart begins to roll until it hits a stump at the bottom of the hill. Upon impact, a 0.25-kg can of peaches flies horizontally out of the shopping cart and hits a parked car with an average force of 500 N. How deep a dent is made in the car (i.e., over what distance does the 500 N force act upon the can of peaches before bringing it to a stop)? Example In the diagram below, 450. joules of work is done raising a 72-newton weight a vertical distance of 5.0 meters. How much work is done to overcome friction as the weight is raised? Example • A box with a mass of 0.04 kg starts from rest at point A and travels 5.00 meters along a uniform track until coming to rest at point B, as shown in the picture. Determine the magnitude of the frictional force acting on the box. (assume the frictional force is constant.) A 0.80 m B 0.50 m example • A block weighing 40. newtons is released from rest on an incline 8.0 meters above the horizontal, as shown in the diagram below. If 50. joules of heat is generated as the block slides down the incline, what is the maximum kinetic energy of the block at the bottom of the incline? The Example of Pendulum Motion • Consider a pendulum bob swinging to and fro on the end of a string. There are only two forces acting upon the pendulum bob. Gravity (an internal force) acts downward and the tensional force (an external force) pulls upwards towards the pivot point. The external force does not do work since at all times it is directed at a 90-degree angle to the motion. The only force doing work is gravity, which is a conservative force. The pendulum • The sum of the kinetic and potential energies in system is called the total ______________________________. mechanical energy • In the case of a pendulum, the total mechanical energy constant (KE + PE) is _________________: at the highest point, all the energy is potential energy, at the lowest point, all the energy is kinetic energy. • As the 2.0-kg pendulum bob in the above diagram swings to and fro, its height and speed change. Use energy equations and the above data to determine the blanks in the above diagram. use the heights and the speeds given in the table below to fill in the remaining cells at the various locations in a 0.200kg bob's trajectory. h (m) Speed (m/s) 2.000 0.0 1.490 1.128 0.897 0.357 PE (J) KE (J) TME (J) 0 3.920 Example • 1. 2. 3. 4. As the pendulum swings from position A to position C as shown in the diagram, what is the relationship of kinetic energy to potential energy? [Neglect friction.] The kinetic energy decreases more than the potential energy increases. The kinetic energy increases more than the potential energy decreases. The kinetic energy decrease is equal to the potential energy increase. The kinetic energy increase is equal to the potential energy decrease. example • A pendulum is pulled to the side and released from rest. Sketch a graph best represents the relationship between the gravitational potential energy of the pendulum and its displacement from its point of release. PE pos Example • 1. 2. 3. 4. In the diagram, an ideal pendulum released from point A swings freely through point B. Compared to the pendulum's kinetic energy at A, its potential energy at B is half as great twice as great the same four times as great Roller coaster – friction is ignored, Wother = 0 • A roller coaster operates on the principle of energy transformation. Work is initially done on a roller coaster car to lift the car to the first and highest hill. The roller coaster car has a large quantity of potential energy and virtually no kinetic energy as it begins the trip down the first hill. As the car descents hills and loops, it potential energy is transformed into kinetic energy; as the car ascends hills and loops, its kinetic energy is transformed into potential energy. The total mechanical energy of the conserved car is _______________ when friction is ignored. • The total mechanical energy of the roller coaster car is a constant value of 40 000 Joules. The skier • Transformation of energy from the potential to the kinetic also occurs for a ski jumper. As a ski jumper glides down the hill towards the jump ramp and off the jump ramp towards the ground, potential energy is transformed into kinetic energy. If friction can be ignored, the total mechanical energy is ______________________. A free falling object • If a stationary object having mass m is located a vertical distance h above Earth’s surface, the object has initial PE = mgh and KE = 0. as object falls, its PE ___________ and KE ________________. The total mechanical energy is conserved. Energy conversion of a free falling object The graph shows as a ball is dropped, how its energy is transformed. • The total mechanical energy remains _____________. • GPE decreases as KE increases example • A 3.0-kilogram object is placed on a frictionless track at point A and released from rest. (Assume the gravitational potential energy of the system to be zero at point C.) Calculate the kinetic energy of the object at point B. example • a. b. c. d. A 250.-kilogram car is initially at rest at point A on a roller coaster track. The car carries a 75-kilogram passenger and is 20. meters above the ground at point A. [Neglect friction.] Compare the total mechanical energy of the car and passenger at points A, B, and C. The total mechanical energy is less at point C than it is at points A or B. The total mechanical energy is greatest at point A. The total mechanical energy is the same at all three points. The total mechanical energy is greatest at point B. example • The diagram represents a 0.20-kilogram sphere moving to the right along a section of a frictionless surface. The speed of the sphere at point A is 3.0 meters per second. • Approximately how much kinetic energy does the sphere gain as it goes from point A to point B? example • A 1.0 kg mass falls freely for 20. meters near the surface of Earth. What is the total KE gained by the object during its free fall? example • Base your answer to the question on the information and diagram. A 250.-kilogram car is initially at rest at point A on a roller coaster track. The car carries a 75-kilogram passenger and is 20. meters above the ground at point A. [Neglect friction.] Calculate the speed of the car and passenger at point B. example • A 20.-kilogram object strikes the ground with 1960 joules of kinetic energy after falling freely from rest. How far above the ground was the object when it was released? Example • A 55.0-kilogram diver falls freely from a diving platform that is 3.00 meters above the surface of the water in a pool. When she is 1.00 meter above the water, what are her gravitational potential energy and kinetic energy with respect to the water's surface? When work is done by elastic force, energy is conserved • A person does 100 joules of work in pulling back the string of a bow. What will be the initial speed of a 0.5-kilogram arrow when it is fired from the bow? example • A vertically hung spring has a spring constant of 150. Newton per meter. A 2.00-kilogram mass is suspended from the spring and allowed to come to rest. Calculate the total elastic potential energy stored in the spring due to the suspended 2.00-kilogram mass. example • The diagram shows a 0.1-kilogram apple attached to a branch of a tree 2 meters above a spring on the ground below. The apple falls and hits the spring, compressing it 0.1 meter from its rest position. If all of the gravitational potential energy of the apple on the tree is transferred to the spring when it is compressed, what is the spring constant of this spring? example • A person does 64 joules of work in pulling back the string of a bow. What will be the initial speed of a 0.5-kilogram arrow when it is fired from the bow? example • A spring in a toy car is compressed a distance, x. When released, the spring returns to its original length, transferring its energy to the car. Consequently, the car having mass m moves with speed v. Derive the spring constant, k, of the car’s spring in terms of m, x, and v. [Assume an ideal mechanical system with no loss of energy.] example • A vertically hung spring has a spring constant of 150. Newton per meter. A 2.00-kilogram mass is suspended from the spring and allowed to come to rest. Calculate the total elastic potential energy stored in the spring due to the suspended 2.00-kilogram mass. example • 1. 2. 3. 4. Which pair of quantities can be expressed using the same units? work and kinetic energy power and momentum impulse and potential energy acceleration and weight example • Which of the following statements are true about work? Include all that apply. 1. Work is a form of energy. 2. Units of work would be equivalent to a Newton times a meter. 3. A kg•m2/s2 would be a unit of work. 4. Work is a time-based quantity; it is dependent upon how fast a force displaces an object. 5. Superman applies a force on a truck to prevent it from moving down a hill. This is an example of work being done. 6. An upward force is applied to a bucket as it is carried 20 m across the yard. This is an example of work being done. 7. A force is applied by a chain to a roller coaster car to carry it up the hill of the first drop of the Shockwave ride. This is an example of work being done. example A B elongation force • Determine the meaning of slope in each graph force weight Gravitational potential energy elongation C D mass height