Subject to much change Chapter 7 Energy and

Subject to much change Chapter 7 Energy and Energy Transfer February 22, 2006

HAPPY BIRTHDAY GEORGE!

Kalendar n n n Today we start the new TOPIC OF ENERGY No Quiz on Friday, but there MAY be on Monday The BAD NEWS: q EXAM #2 will be on March 3 (Friday)

ENERGY n n n We use energy to walk, run or even sleep We use energy when we lift a weight We use energy when we drive a car We even use energy to THINK! BUT …. .

Introduction to Energy The concept of energy is one of the most important topics in science n Every physical process that occurs in the Universe involves energy and energy transfers or transformations n Energy is not easily defined n

Systems n A system is a small portion of the Universe q q q n We identify a number of particles or objects and draw a sphere around them There are no forces acting on anything inside the sphere from outside the sphere We will ignore the details outside of the sphere. A critical skill is to identify the system

Valid System n A valid system may q q be a single object or particle be a collection of objects or particles be a region of space vary in size and shape

Environment n There is a system boundary around the system q q n The boundary is an imaginary surface It does not necessarily correspond to a physical boundary The boundary divides the system from the environment q The environment is “the rest of the Universe”

Work n The work, W, done on a system by an agent exerting a constant force on the system is the product of the magnitude, F, of the force, the magnitude Dr of the displacement of the point of application of the force, and cos q, where q is the angle between the force and the displacement vectors

Working, Working WORK = Component of the applied force x the displacement= Fcos(q) x Dr

Work, cont. n W = F Dr cos q q (Later for the dot!) The displacement is that of the point of application of the force 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

Work Example n The normal force, n, and the gravitational force, m g, do no work on the object q n n cos q = cos 90° = 0 The force F does do work on the object Same amount as in the previous overhead

More About Work n The system and the environment must be determined when dealing with work q The environment does work on the system n n Work by the environment on the system The sign of the work depends on the direction of F relative to Dr q q Work is positive when projection of F onto Dr is in the same direction as the displacement Work is negative when the projection is in the opposite direction

Units of Work n n Work is a scalar quantity The unit of work is a joule (J) q q 1 joule = 1 newton. 1 meter J=N·m

A block of mass 2. 50 kg is pushed 2. 20 m along a frictionless horizontal table by a constant 16. 0 -N force directed 25. 0 below the horizontal. Determine the work done on the block by (a) the applied force, (b) the normal force exerted by the table, and (c) the gravitational force. (d) Determine the total work done on the block.

A raindrop of mass 3. 35 10– 5 kg falls vertically at constant speed under the influence of gravity and air resistance. Model the drop as a particle. As it falls 100 m, what is the work done on the raindrop (a) by the gravitational force and (b) by air resistance?

Work Is An Energy Transfer n n n This is important for a system approach to solving a problem If the work is done on a system and it is positive, energy is transferred to the system If the work done on the system is negative, energy is transferred from the system

Work Is An Energy Transfer, cont n If a system interacts with its environment, this interaction can be described as a transfer of energy across the system boundary q This will result in a change in the amount of energy stored in the system

Shejule Continue to work on energy. n Exam on March 3 rd. n Material … as far as we get by March 1 st. n Mucho Web. Assign Stuff n

LAST TIME We defined the “Dot Product” n We defined WORK n We discussed the “system” and the “environment” n System + Environment = Entire Universe (Dumb concept!) n

WORK F is the NET force acting on the block.

Scalar or DOT Product of Two Vectors n The scalar product of two vectors is written as A. B q n It is also called the dot product A. B = A B cos q q q is the angle between A and B

Scalar Product Properties n The scalar product is commutative q n n A. B = B. A The scalar product obeys the distributive law of multiplication A. (B + C) = A. B + A. C

Dot Products of Unit Vectors n n Using component form with A and B:

A force acts on a particle that undergoes a displacement. Find (a) the work done by the force on the particle and (b) the angle between F and r.

Work Done by a Varying Force

11. The force acting on a particle varies as in Figure P 7. 11. Find the work done by the force on the particle as it moves (a) from x = 0 to x = 8. 00 m, (b) from x = 8. 00 m to x = 10. 0 m, and (c) from x = 0 to x = 10. 0 m.

Work Done By Multiple Forces n AGAIN … If more than one force acts on a system and the system can be modeled as a particle, the total work done on the system is the work done by the net force

Hooke’s Law n The force exerted by the spring is Fs = - kx q q n x is the position of the block with respect to the equilibrium position (x = 0) k is called the spring constant or force constant and measures the stiffness of the spring This is called Hooke’s Law

Hooke’s Law, cont. n n n When x is positive (spring is stretched), F is negative When x is 0 (at the equilibrium position), F is 0 When x is negative (spring is compressed), F is positive

Work Done by a Spring n n n Identify the block as the system Calculate the work as the block moves from xi = xmax to xf = 0 The total work done as the block moves from –xmax to xmax is zero ENERGY IS STORED IN THE SPRING AND THEN RECOVERED AND THEN STORED AND THEN RECOVERED AND THEN ….

Spring with an Applied Force n n Suppose an external agent, Fapp, stretches the spring The applied force is equal and opposite to the spring force Fapp = -Fs = -(-kx) = kx Work done by Fapp is equal to ½ kx 2 max

19. If it takes 4. 00 J of work to stretch a Hooke's-law spring 10. 0 cm from its unstressed length, determine the extra work required to stretch it an additional 10. 0 cm.

21. A light spring with spring constant 1 200 N/m is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant 1 800 N/m. An object of mass 1. 50 kg is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.

Consider the following: F

Kinetic Energy n Kinetic Energy is the energy of a particle due to its motion q K = ½ mv 2 n n K is the kinetic energy m is the mass of the particle v is the speed of the particle A change in kinetic energy is one possible result of doing work to transfer energy into a system With no friction or other strange forces, the work done by a force on a particle (or system of particles) = the change in the particles kinetic energy.

Work-Kinetic Energy Theorem n n n The Work-Kinetic Energy Principle states SW = Kf – Ki = DK In the case in which work is done on a system and the only change in the system is in its speed, the work done by the net force equals the change in kinetic energy of the system. We can also define the kinetic energy q K = ½ mv 2

Nonisolated System n A nonisolated system is one that interacts with or is influenced by its environment q n An isolated system would not interact with its environment The Work-Kinetic Energy Theorem can be applied to nonisolated systems

BREAK POINT February 27, 2006

Stuff Happens n Today and Wednesday – More on potential energy – some material from the next chapter n Friday – EXAMINATION #2 – All material since last exam. – You should be studying by now.

Internal Energy n n n The energy associated with an object’s temperature is called its internal energy, Eint In this example, the surface is the system The friction does work and increases the internal energy of the surface FRICTION

Potential Energy n n Potential energy is energy related to the configuration of a system in which the components of the system interact by forces Examples include: q q q elastic potential energy – stored in a spring gravitational potential energy electrical potential energy

Conservation of Energy n Energy is conserved !!!! q q This means that energy cannot be created or destroyed If the total amount of energy in a system changes, it can only be due to the fact that energy has crossed the boundary of the system by some method of energy transfer

A 2 100 -kg pile driver is used to drive a steel I-beam into the ground. The pile driver falls 5. 00 m before coming into contact with the top of the beam, and it drives the beam 12. 0 cm farther into the ground before coming to rest. Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest.

31. A 40. 0 -kg box initially at rest is pushed 5. 00 m along a rough, horizontal floor with a constant applied horizontal force of 130 N. If the coefficient of friction between box and floor is 0. 300, find (a) the work done by the applied force, (b) the increase in internal energy in the box-floor system due to friction, (c) the work done by the normal force, (d) the work done by the gravitational force, (e) the change in kinetic energy of the box, and (f) the final speed of the box.

63. The ball launcher in a pinball machine has a spring that has a force constant of 1. 20 N/cm (Fig. P 7. 63). The surface on which the ball moves is inclined 10. 0° with respect to the horizontal. If the spring is initially compressed 5. 00 cm, find the launching speed of a 100 -g ball when the plunger is released. Friction and the mass of the plunger are negligible.

32. A 2. 00 -kg block is attached to a spring of force constant 500 N/m as in Figure 7. 10. The block is pulled 5. 00 cm to the right of equilibrium and released from rest. Find the speed of the block as it passes through equilibrium if (a) the horizontal surface is frictionless and (b) the coefficient of friction between block and surface is 0. 350.

35. A sled of mass m is given a kick on a frozen pond. The kick imparts to it an initial speed of 2. 00 m/s. The coefficient of kinetic friction between sled and ice is 0. 100. Use energy considerations to find the distance the sled moves before it stops.

Power n n The time rate of energy transfer is called power The average power is given by when the method of energy transfer is work

Instantaneous Power n The instantaneous power is the limiting value of the average power as Dt approaches zero n This can also be written as

Power Generalized n n n Power can be related to any type of energy transfer In general, power can be expressed as d. E/dt is the rate at which energy is crossing the boundary of the system for a given transfer mechanism

Units of Power n The SI unit of power is called the watt q n A unit of power in the US Customary system is horsepower q n 1 watt = 1 joule / second = 1 kg. m 2 / s 2 1 hp = 746 W Units of power can also be used to express units of work or energy q 1 k. Wh = (1000 W)(3600 s) = 3. 6 x 106 J

49. A 4. 00 -kg particle moves along the x axis. Its position varies with time according to x = t + 2. 0 t 3, where x is in meters and t is in seconds. Find (a) the kinetic energy at any time t, (b) the acceleration of the particle and the force acting on it at time t, (c) the power being delivered to the particle at time t, and (d) the work done on the particle in the interval t = 0 to t = 2. 00 s.

BREAK

EXAM #2 This Friday!

Let’s Review with some Problems

24. A 5. 00 -kg object placed on a frictionless, horizontal table is connected to a cable that passes over a pulley and then is fastened to a hanging 9. 00 -kg object, as in Figure P 5. 24. Draw free-body diagrams of both objects. Find the acceleration of the two objects and the tension in the string.

In Figure P 5. 29, the man and the platform together weigh 950 N. The pulley can be modeled as frictionless. Determine how hard the man has to pull on the rope to lift himself steadily upward above the ground. (Or is it impossible? If so, explain why. )

34. An object of mass m 1 on a frictionless horizontal table is connected to an object of mass m 2 through a very light pulley P 1 and a light fixed pulley P 2 as shown in Figure P 5. 34. (a) If a 1 and a 2 are the accelerations of m 1 and m 2, respectively, what is the relation between these accelerations? Express (b) the tensions in the strings and (c) the accelerations a 1 and a 2 in terms of the masses m 1, and m 2, and g.

44. Three objects are connected on the table as shown in Figure P 5. 44. The table is rough and has a coefficient of kinetic friction of 0. 350. The objects have masses 4. 00 kg, 1. 00 kg and 2. 00 kg, as shown, and the pulleys are frictionless. Draw free-body diagrams of each of the objects. (a) Determine the acceleration of each object and their directions. (b) Determine the tensions in the two cords.

68. Two blocks of mass 3. 50 kg and 8. 00 kg are connected by a massless string that passes over a frictionless pulley (Fig. P 5. 68). The inclines are frictionless. Find (a) the magnitude of the acceleration of each block and (b) the tension in the string.

19. A roller coaster car (Fig. P 6. 19) has a mass of 500 kg when fully loaded with passengers. (a) If the vehicle has a speed of 20. 0 m/s at point A, what is the force exerted by the track on the car at this point? (b) What is the maximum speed the vehicle can have at B and still remain on the track?

25. A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591 N. As the elevator later is stopping, the scale reading is 391 N. Assume the magnitude of the acceleration is the same during starting and stopping, and determine (a) the weight of the person, (b) the person's mass, and (c) the acceleration of the elevator.

7. A crate of eggs is located in the middle of the flat bed of a pickup truck as the truck negotiates an unbanked curve in the road. The curve may be regarded as an arc of a circle of radius 35. 0 m. If the coefficient of static friction between crate and truck is 0. 600, how fast can the truck be moving without the crate sliding?

8. Find the scalar product of the vectors in Figure P 7. 8.

15. When a 4. 00 -kg object is hung vertically on a certain light spring that obeys Hooke's law, the spring stretches 2. 50 cm. If the 4. 00 -kg object is removed, (a) how far will the spring stretch if a 1. 50 -kg block is hung on it, and (b) how much work must an external agent do to stretch the same spring 4. 00 cm from its unstretched position?

20. A small particle of mass m is pulled to the top of a frictionless halfcylinder (of radius R) by a cord that passes over the top of the cylinder, as illustrated in Figure P 7. 20. (a) If the particle moves at a constant speed, show that F = mgcos. (Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times. ) (b) By directly integrating W = F dr, find the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder.

49. A 4. 00 -kg particle moves along the x axis. Its position varies with time according to x = t + 2. 0 t 3, where x is in meters and t is in seconds. Find (a) the kinetic energy at any time t, (b) the acceleration of the particle and the force acting on it at time t, (c) the power being delivered to the particle at time t, and (d) the work done on the particle in the interval t = 0 to t = 2. 00 s.

Potential Energy ~March 7, 2006

Potential (Stored) Energy n Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other q This can be used only with conservative forces n q Conservative forces are NOT Republicans When conservative forces act within an isolated system, the kinetic energy gained (or lost) by the system as its members change their relative positions is balanced by an equal loss (or gain) in potential energy. n This is Conservation of Mechanical Energy

Types of Potential Energy n n n There are many forms of potential energy, including: q Gravitational q Electromagnetic q Chemical q Nuclear One form of energy in a system can be converted into another q Nuclear heat easy q Heat Nuclear probably impossible! Conversion from one type to another type of energy is not always reversible.

Systems with Multiple Particles n n n We can extend our definition of a system to include multiple objects The force can be internal to the system The kinetic energy of the system is the algebraic sum of the kinetic energies of the individual objects q Sometimes, the kinetic energy of one of the objects may be negligible

System Example n n n n This system consists of Earth and a book Do work on the system by lifting the book through Dy The work done by you is mg(yb – ya) At the top it is at rest. The amount of work that you did is called the potential energy of the system with respect to the ground. The PE’s initial value is mgya The FINAL value is mgyb The difference is the work done.

Let’s drop the book from yb and see what it is doing at ya. Multiply by m and divide by 2: The decrease in Potential Energy equals the increase in Kinetic Energy

Potential Energy n n n The energy storage mechanism is called potential energy A potential energy can only be associated with specific types of forces (conservative) Potential energy is always associated with a system of two or more interacting objects

Gravitational Potential Energy n n n Gravitational Potential Energy is associated with an object at a given distance above Earth’s surface Assume the object is in equilibrium and moving at constant velocity The work done on the object is done by Fapp and the upward displacement is

Gravitational Potential Energy, cont n n The quantity mgy is identified as the gravitational potential energy, Ug q n Ug = mgy Units are joules (J)

Energy Problems n n n Draw a diagram of the situation. ESTABLISH AN ORIGIN. THE POTENTIAL ENERGY OF A PARTICAL OF MASS M IS ALWAYS MEASURED WITH RESPECT TO THIS ORIGIN. The potential energy of a particle is defined as being ZERO when it is at the origin. At some height above the origin, the value of the PE is mgh.

Gravitational Potential Energy, final n n The gravitational potential energy depends only on the vertical height of the object above Earth’s surface In solving problems, you must choose a reference configuration for which the gravitational potential energy is set equal to some reference value, normally zero q The choice is arbitrary because you normally need the difference in potential energy, which is independent of the choice of reference configuration

Conservation of Mechanical Energy n The mechanical energy of a system is the algebraic sum of the kinetic and potential energies in the system q n Emech = K + Ug The statement of Conservation of Mechanical Energy for an isolated system is Kf + Uf = Ki+ Ui q An isolated system is one for which there are no energy transfers across the boundary

A bead slides without friction around a loop-the-loop (Fig. P 8. 5). The bead is released from a height h = 3. 50 R. (a) What is its speed at point A? (b) How large is the normal force on it if its mass is 5. 00 g?

Let’s look at the more general case. y=h m W=0 d D F y=0 (origin) W=mg. D We do work to move mass to y=h. W=mgh m m

ANY PATH n n Can be broken up into a series of very small vertical moves and horizontal moves. The horizontal moves require no work. q n The force is at right angles to the motion. Dot product is zero. The vertical moves are

Conservation of Mechanical Energy, example n n n Look at the work done by the book as it falls from some height to a lower height Won book = DKbook Also, W = mgyb – mgya So, DK = -Dug Or: DK +Dug=0

Elastic Potential Energy n n n Elastic Potential Energy is associated with a spring The force the spring exerts (on a block, for example) is Fs = - kx The work done by an external applied force on a spring-block system is q q W = ½ kxf 2 – ½ kxi 2 The work is equal to the difference between the initial and final values of an expression related to the configuration of the system

Elastic Potential Energy, cont n n n This expression is the elastic potential energy: Us = ½ kx 2 The elastic potential energy can be thought of as the energy stored in the deformed spring The stored potential energy can be converted into kinetic energy

Elastic Potential Energy, final n The elastic potential energy stored in a spring is zero whenever the spring is not deformed (U = 0 when x = 0) q n n The energy is stored in the spring only when the spring is stretched or compressed The elastic potential energy is a maximum when the spring has reached its maximum extension or compression The elastic potential energy is always positive q x 2 will always be positive

The Bindell Conservation of Energy Equation The sum of the Kinetic Energy and the potential energy of the system before a sequence of events Is Equal to The sum of the Kinetic Energy, the potential energy of the system and the Energy Lost to friction or witchcraft.

An object of mass m starts from rest and slides a distance d down a frictionless incline of angle. While sliding, it contacts an unstressed spring of negligible mass as shown in Figure P 8. 10. The object slides an additional distance x as it is brought momentarily to rest by compression of the spring (of force constant k). Find the initial separation d between object and spring. CAN WE ADD FRICTION? ? ?

Problem Solving Strategy – Conservation of Mechanical Energy n Define the isolated system and the initial and final configuration of the system q q q The system may include two or more interacting particles The system may also include springs or other structures in which elastic potential energy can be stored Also include all components of the system that exert forces on each other

Problem-Solving Strategy, 2 n Identify the configuration for zero potential energy q q Include both gravitational and elastic potential energies If more than one force is acting within the system, write an expression for the potential energy associated with each force

Problem-Solving Strategy, 3 n n If friction or air resistance is present, mechanical energy of the system is not conserved Use energy with non-conservative forces instead

Problem-Solving Strategy, 4 n If the mechanical energy of the system is conserved, write the total energy as q q n Ei = Ki + Ui for the initial configuration Ef = Kf + Uf for the final configuration Since mechanical energy is conserved, Ei = Ef and you can solve for the unknown quantity

Two objects are connected by a light string passing over a light frictionless pulley as in Figure P 8. 13. The object of mass m 1 is released from rest at height h. Using the principle of conservation of energy, (a) determine the speed of m 2 just as m 1 hits the ground. (b) Find the maximum height to which m 2 rises.

A particle of mass m = 5. 00 kg is released from point A and slides on the frictionless track shown in Figure P 8. 24. Determine (a) the particle's speed at points B and C and (b) the net work done by the gravitational force in moving the particle from A to C.

A block slides down a curved frictionless track and then up an inclined plane as in Figure P 8. 48. The coefficient of kinetic friction between block and incline is k. Use energy methods to show that the maximum height reached by the block is

A 10. 0 -kg block is released from point A in Figure P 8. 57. The track is frictionless except for the portion between points B and C , which has a length of 6. 00 m. The block travels down the track, hits a spring of force constant 2 250 N/m, and compresses the spring 0. 300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between B and C.

Conservation of Energy, (Pendulum) n n As the pendulum swings, there is a continuous change between potential and kinetic energies At A, the energy is potential At B, all of the potential energy at A is transformed into kinetic energy q Let zero potential energy be at B At C, the kinetic energy has been transformed back into potential energy

Conservative Forces n n The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle The work done by a conservative force on a particle moving through any closed path is zero q A closed path is one in which the beginning and ending points are the same

Nonconservative Forces n n A nonconservative force does not satisfy the conditions of conservative forces Nonconservative forces acting in a system cause a change in the mechanical energy of the system

Nonconservative Forces n n The work done against friction is greater along the red path than along the blue path Because the work done depends on the path, friction is a nonconservative force

Nonconservative Forces (Connected Blocks) n n n The system consists of the two blocks, the spring, and Earth Gravitational and potential energies are involved The kinetic energy is zero if our initial and final configurations are at rest

Connected Blocks, cont n n n Block 2 undergoes a change in gravitational potential energy The spring undergoes a change in elastic potential energy The coefficient of kinetic energy can be measured

Conservative Forces and Potential Energy n n Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system The work done by such a force, F, is q DU is negative when F and x are in the same direction

Conservative Forces and Potential Energy n The conservative force is related to the potential energy function through n The x component of a conservative force acting on an object within a system equals the negative of the potential energy of the system with respect to x

Conservative Forces and Potential Energy – Check n Look at the case of a deformed spring q This is Hooke’s Law

Energy Diagrams and Equilibrium n n n Motion in a system can be observed in terms of a graph of its position and energy In a spring-mass system example, the block oscillates between the turning points, x = ±xmax The block will always accelerate back toward x = 0

Energy Diagrams and Stable Equilibrium n n n The x = 0 position is one of stable equilibrium Configurations of stable equilibrium correspond to those for which U(x) is a minimum x=xmax and x=-xmax are called the turning points

Energy Diagrams and Unstable Equilibrium n n Fx = 0 at x = 0, so the particle is in equilibrium For any other value of x, the particle moves away from the equilibrium position This is an example of unstable equilibrium Configurations of unstable equilibrium correspond to those for which U(x) is a maximum