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Work and Energy Chin-Sung Lin Work and Energy Chin-Sung Lin

Energy A central concept underlying all sciences However, concept of energy is very difficult Energy A central concept underlying all sciences However, concept of energy is very difficult to define Concept of energy is unknown to Isaac Newton. Its existence was still debated in the 1850 s A quantity that is often understood as the capacity to do work on a physical system

Energy We observe only the effects of energy when something is happening When energy Energy We observe only the effects of energy when something is happening When energy is being transferred from one place to another When energy is being transformed from one form to another

Forms of Energy Thermal energy Elastic energy Chemical energy Sound Energy Electrical energy Mechanical Forms of Energy Thermal energy Elastic energy Chemical energy Sound Energy Electrical energy Mechanical energy Radiant energy Luminous energy Nuclear energy Magnetic energy

Work The amount of energy transferred by a force acting through a distance Work Work The amount of energy transferred by a force acting through a distance Work = net force x distance W = Fnet x d Unit of work: newton-meter (N m) = joule (J)

Work A scalar quantity Independent of time Unit of work: newton-meter (N m) = Work A scalar quantity Independent of time Unit of work: newton-meter (N m) = joule (J) 1 J = 1 N x 1 m = 1 kg m 2/s 2 1 KJ = 1, 000 J 1 MJ = 1, 000 J

Work Work

Work Work

Work Two ways to do work Force acts to change motion Force acts against Work Two ways to do work Force acts to change motion Force acts against another force

Work Example An 80 N force has been applied to a block and move Work Example An 80 N force has been applied to a block and move it 20 m along the direction of the force. How much work has been done to the block?

Work Example An 80 N force has been applied to a block and move Work Example An 80 N force has been applied to a block and move it 20 m along the direction of the force. How much work has been done to the block? W=Fd = 80 N x 20 m = 1600 J

Work Exercise Calculate the work done when a 20 -N force pushes a cart Work Exercise Calculate the work done when a 20 -N force pushes a cart 3. 5 m ?

Work Exercise How much work is required to lift a 360 kilogram piano to Work Exercise How much work is required to lift a 360 kilogram piano to a window whose height is 10 meters from the ground?

Work Exercise A box rests on a horizontal, frictionless surface. A girl pushes on Work Exercise A box rests on a horizontal, frictionless surface. A girl pushes on the box with a force of 18 N to the right and a boy pushes on the box with a force of 12 N to the left. The box moves 4. 0 m to the right. Find the work done by (a) the girl, (b) the boy, and (c) the net force.

Work vs. Impulse impulse = net force x time J=Ft impulse changes the momentum Work vs. Impulse impulse = net force x time J=Ft impulse changes the momentum of a system unit of impulse: newton-second (N s) work = net force x distance W=Fd work changes the energy of a system unit of work: newton-meter (N m) = joule (J)

Work and Force with an Angle W = F cos θ d d Work and Force with an Angle W = F cos θ d d

Work Exercise If 100 N force has 30 o angle pulling on 15 kg Work Exercise If 100 N force has 30 o angle pulling on 15 kg block for 5 m. What’s the work?

Work Exercise If 100 N force has 30 o angle pulling on 15 kg Work Exercise If 100 N force has 30 o angle pulling on 15 kg block for 5 m. What’s the work? W = F d cos W = 100 N (5 m) cos 30 o = 433 J

Positive or Negative Work Man does positive work lifting box Man does negative work Positive or Negative Work 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

Power – The rate at which work is done Power = (Work Done) / Power – The rate at which work is done Power = (Work Done) / (Time Interval) P = W/t Unit of power: joule per second , or Watt (W) 1 watt (W) = 1 J / 1 second 1 kilowatt (k. W) = 1, 000 watts 1 megawatt (MW) = 1, 000 watts

Power When W is in the equation, W means work; when W is in Power When W is in the equation, W means work; when W is in the unit, W means Watt In metric system of units, automobiles are rated in kilowatts; in the US, we rate engines in units of horsepower (hp). 1 hp = 0. 75 k. W or 134 hp = 100 k. W

Power Example Calculate the power expended when a 500 N barbell is lifted 2. Power Example Calculate the power expended when a 500 N barbell is lifted 2. 2 m in 2 s

Power Example Calculate the power expended when a 500 N barbell is lifted 2. Power Example Calculate the power expended when a 500 N barbell is lifted 2. 2 m in 2 s W = Fd/t = 500 N x 2. 2 m / 2 s = 1, 100 J / 2 s = 550 watts

Power Exercise An escalator is used to move 20 passengers every minute from the Power Exercise An escalator is used to move 20 passengers every minute from the first floor of a department store to the second. The second floor is located 5 -meters above the first floor. The average passenger's mass is 60 kg. Determine the power requirement of the escalator in order to move this number of passengers in this amount of time

Power = Force x Average Velocity P = W/t = Fd/t = Fv Power = Force x Average Velocity P = W/t = Fd/t = Fv

Power Example A 500 N force is applied to an object. If the object Power Example A 500 N force is applied to an object. If the object travels with a constant velocity of 20 meters per second, calculate the power expended on the object

Power Example A 500 N force is applied to an object. If the object Power Example A 500 N force is applied to an object. If the object travels with a constant velocity of 20 meters per second, calculate the power expended on the object P = Fv = (500 N)(20 m/s) = 10, 000 W = 10 k. W

Power Exercise An elevator must lift 1000 kg a distance of 100 m at Power Exercise An elevator must lift 1000 kg a distance of 100 m at a velocity of 4 m/s. What is the average power the elevator exerts during this trip?

Mechanical Energy A compression of atoms in the material of an object, a physical Mechanical Energy A compression of atoms in the material of an object, a physical separation of attracting bodies, or a rearrangement of electric charges in the molecules of a substance Energy enables an object to do work Unit of energy: energy is measured in joules

Mechanical Energy Two most common forms of mechanical energy: potential energy (PE) and kinetic Mechanical Energy Two most common forms of mechanical energy: potential energy (PE) and kinetic energy (KE) Mechanical energy can be in the form of either one, or the sum of the two Potential energy— the energy due to the position of something, or Kinetic energy— the energy due to the movement of something

Potential Energy Definition: An object may store energy by virtue of its position The Potential Energy Definition: An object may store energy by virtue of its position The energy that is held in readiness is called potential energy (PE), because in the stored state it has the potential for doing work

Potential Energy Examples of potential energy: A stretched/compressed spring A drawn bow A stretched Potential Energy Examples of potential energy: A stretched/compressed spring A drawn bow A stretched rubber band The chemical energy in fuels (at submicroscopic level) Any substance that can do work through chemical action possesses potential energy (such as fossil fuels, electric batteries and food)

Gravitational Potential Energy Work is required to elevate objects against Earth’s gravity The potential Gravitational Potential Energy Work is required to elevate objects against Earth’s gravity The potential energy due to elevated positions is called gravitational potential energy (GPE) The amount of gravitational potential energy possessed by an elevated object is equal to the work done against gravity in lifting it

Gravitational Potential Energy The upward force required while moving at constant velocity is equal Gravitational Potential Energy The upward force required while moving at constant velocity is equal to the weight (mg) of the object Gravitational Potential Energy = Weight x Height GPE = Fgh = mgh or PE = mgh The gravitation potential energy depends only on mg and h, does not depend on the path Unit of gravitational potential energy: kg m 2/s 2 = joule (J)

Potential Energy Example The floor of the basement of a house is 3. 0 Potential Energy Example The floor of the basement of a house is 3. 0 m below ground level, and the floor of the attic is 5. 0 m above ground level. What is the change in potential energy if an 2. 0 kg object in the attic is brought to the basement?

Potential Energy Example The floor of the basement of a house is 3. 0 Potential Energy Example The floor of the basement of a house is 3. 0 m below ground level, and the floor of the attic is 5. 0 m above ground level. What is the change in potential energy if an 2. 0 kg object in the attic is brought to the basement? PE = mgh = (2. 0 kg)(9. 8 m/s 2)(5. 0 m + 3. 0 m) = 156. 8 J

Potential Energy Exercise A box has a mass of 5. 8 kg. The box Potential Energy Exercise A box has a mass of 5. 8 kg. The box is lifted from the garage floor and placed on a shelf. If the box gains 145 J of Potential Energy (Ep), how high is the shelf?

Potential Energy Exercise A man climbs on to a wall that is 3. 6 Potential Energy Exercise A man climbs on to a wall that is 3. 6 m high and gains 2268 J of potential energy. What is the mass of the man?

Spring Force Hooke’s Law: Hooke's Law says that the spring restoration force (FS) due Spring Force Hooke’s Law: Hooke's Law says that the spring restoration force (FS) due to a spring is proportional to the length that the spring is stretched or compressed, and acts in the opposite direction FS = kx k is spring constant, and x change in spring length from the equilibrium position Unit of spring force: newton (N)

Hooke’s Law The more force that was put on materials the more they extended Hooke’s Law The more force that was put on materials the more they extended With some materials they also extended in a regular way e. g. if the force was doubled so did the extension This was true as long as their elastic limit was not exceeded

Hooke’s Law Hooke’s Law

Force vs. Extension Force vs. Extension

Elastic limit The material no longer shows elastic behavior (i. e. does not return Elastic limit The material no longer shows elastic behavior (i. e. does not return to original size when stretching force is removed) The material is permanently deformed i. e. , is larger or longer than originally

Spring Constant k is called the spring constant and is a measure of the Spring Constant k is called the spring constant and is a measure of the stiffness of the spring or material Unit of of k: N/m (newtons per meter) The higher the k the stiffer the spring, it needs a large force to for a given extension

Spring Force Example A linear spring has a constant of 377. 16 N/m. How Spring Force Example A linear spring has a constant of 377. 16 N/m. How much force is necessary to stretch it 0. 39 m?

Spring Force Example A linear spring has a constant of 377. 16 N/m. How Spring Force Example A linear spring has a constant of 377. 16 N/m. How much force is necessary to stretch it 0. 39 m? Fs = kx = (377. 16 N/m)(0. 39 m) = 147. 09 N

Spring Force Exercise It requires a force of 594. 55 N to stretch a Spring Force Exercise It requires a force of 594. 55 N to stretch a certain linear spring 0. 15 m. What is the constant for this spring?

Spring Potential Energy Spring Potential Energy: A spring may store energy due to compression Spring Potential Energy Spring Potential Energy: A spring may store energy due to compression or stretching and is called spring potential energy (PES) The potential energy stored in spring is equal to the work done to the spring: PES = FSx = (1/2 FS)x = (1/2 kx)x = 1/2 kx 2 PES = 1/2 kx 2 Unit of Spring Potential Energy: N m = joule (J)

Spring Potential Energy Example Vincent's mountain bike has a spring with a constant of Spring Potential Energy Example Vincent's mountain bike has a spring with a constant of 64 N/m in the front-wheel suspension, and it compressed 0. 17 m when she hit a bump. How much energy does the front spring now store?

Spring Potential Energy Example Vincent's mountain bike has a spring with a constant of Spring Potential Energy Example Vincent's mountain bike has a spring with a constant of 64 N/m in the front-wheel suspension, and it compressed 0. 17 m when she hit a bump. How much energy does the front spring now store? PEs = (1/2) kx 2 = (1/2)(64 N/m)(0. 17 m)2 = 0. 92 J

Spring Potential Energy Exercise A spring has a potential energy of 22. 14 J Spring Potential Energy Exercise A spring has a potential energy of 22. 14 J and a constant of 676. 96 N/m. How far has it been stretched?

Kinetic Energy If an object is moving, then it is capable of doing work. Kinetic Energy If an object is moving, then it is capable of doing work. It has energy of motion, or kinetic energy (KE) The kinetic energy of a moving object is equal to the work required to bring it to that speed from the rest, or the work the object can do while being brought to rest The kinetic energy of an object depends on the mass and speed of the object

Kinetic Energy Work = Net Force x Distance = Kinetic Energy KE = W Kinetic Energy Work = Net Force x Distance = Kinetic Energy KE = W = Fd = mad = m(v 2/(2 d))d = 1/2 mv 2 KE = 1/2 mv 2 Unit of kinetic energy: kg m 2/s 2 = joule (J)

Kinetic Energy Example A 3 kg ball is rolling 2 m/s. How much kinetic Kinetic Energy Example A 3 kg ball is rolling 2 m/s. How much kinetic energy does it have?

Kinetic Energy Example A 3 kg ball is rolling 2 m/s. How much kinetic Kinetic Energy Example A 3 kg ball is rolling 2 m/s. How much kinetic energy does it have? KE = ½ mv 2 = ½ (3 kg) (2 m/s)2 = 6 J

Kinetic Energy Exercise Determine the kinetic energy of a 500 -kg roller coaster car Kinetic Energy Exercise Determine the kinetic energy of a 500 -kg roller coaster car that is moving with a speed of 20 m/s If the roller coaster car were moving with twice the speed, then what would be its new kinetic energy?

Kinetic Energy Exercise Missy Diwater, the former platform diver for the Ringling Brother's Circus, Kinetic Energy Exercise Missy Diwater, the former platform diver for the Ringling Brother's Circus, had a kinetic energy of 12000 J just prior to hitting the bucket of water. If Missy's mass is 40 kg, then what is her speed?

Kinetic Energy If the speed of an object is doubled (the speed is squared), Kinetic Energy If the speed of an object is doubled (the speed is squared), its kinetic energy is quadrupled It takes four times the work to double the speed It takes four times of work to stop a double-speed object Work-Energy Theorem: Work changes energy. If no change in energy occurs, then no work is done. W= E

Work-Energy Theorem Example A skater of mass 60 kg has an initial velocity of Work-Energy Theorem Example A skater of mass 60 kg has an initial velocity of 12 m/s. He slides on ice where the frictional force is 36 N. How far will the skater slide before he stops?

Work-Energy Theorem Example A skater of mass 60 kg has an initial velocity of Work-Energy Theorem Example A skater of mass 60 kg has an initial velocity of 12 m/s. He slides on ice where the frictional force is 36 N. How far will the skater slide before he stops? W = E = KE Fnet d = 1/2 mv 2 36 N x d = ½ (60 kg)(12 m/s)2 d = 120 m

Work Exercise When a small brass ball is dropped into soft clay, it makes Work Exercise When a small brass ball is dropped into soft clay, it makes a dent. If the ball hits with twice the speed, the dent of the clay is a) more or less the same depth. b) twice as deep. c) three times as deep d) four times as deep.

Conservation of Energy Law of conservation of energy: Energy cannot be created or destroyed. Conservation of Energy Law of conservation of energy: Energy cannot be created or destroyed. It can be transformed from one form into another, but the total amount of energy never changes Total Energy at point A = Total Energy at point B PEA + KEA = PEB + KEB PE = mgh KE = 1/2 mv 2

Conservation of Energy Example A diver of mass m drops from a board 10. Conservation of Energy Example A diver of mass m drops from a board 10. 0 m above the water surface. Find his speed 5. 00 m above the water surface

Conservation of Energy Example A diver of mass m drops from a board 10. Conservation of Energy Example A diver of mass m drops from a board 10. 0 m above the water surface. Find his speed 5. 00 m above the water surface PE = KE mg h = ½ m v 2 m(9. 81 m/s)(10 m – 5 m) = ½ m v 2 v = 9. 9 m/s

Conservation of Energy Exercise A diver of mass m drops from a board 10. Conservation of Energy Exercise A diver of mass m drops from a board 10. 0 m above the water surface. Find his speed right above the water surface

Conservation of Energy Exercise A skier slides down the frictionless slope as shown. What Conservation of Energy Exercise 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

Conservation of Energy Exercise Three identical balls are thrown from the top of a Conservation of Energy Exercise Three identical balls are thrown from the top of a building with the same initial speed. Initially, Ball 1 moves horizontally. Ball 2 moves upward. Ball 3 moves downward. Neglecting air resistance, which ball has the fastest speed when it hits the ground? A) Ball 1 B) Ball 2 C) Ball 3 D) All the same

Conservation of Energy Exercise Tarzan swings from a vine whose length is 12 m. Conservation of Energy Exercise Tarzan swings from a vine whose length is 12 m. If Tarzan starts at an angle of 30 degrees with respect to the vertical and has no initial speed, what is his speed at the bottom of the arc?