Chapter 8 Potential Energy & Conservation of Energy

Potential Energy Potential energy is energy related to the configuration of a system in which the components of the system interact by forces The forces are internal to the system Can be associated with only specific types of forces acting between members of a system

Gravitational Potential Energy The system is the Earth and the book Do work on the book by lifting it slowly through a vertical displacement The work done on the system The work done on the system must appear as an increase in the energy of the system

Gravitational Potential Energy, cont There is no change in kinetic energy since the book starts and ends at rest Gravitational potential energy is the energy associated with an object at a given location above the surface of the Earth

Gravitational Potential Energy, final The quantity mgy is identified as the gravitational potential energy, Ug U g = mgy Units are joules (J) Is a scalar Work may change the gravitational potential energy of the system W net =  U g

Gravitational Potential Energy, Problem Solving 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 The choice is arbitrary because you normally need the difference in potential energy, which is independent of the choice of reference configuration

Elastic Potential Energy Elastic Potential Energy is associated with a spring The force the spring exerts (on a block, for example) is F s = - kx The work done by an external applied force on a spring-block system is W = ½ kx f 2 – ½ kx i 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 This expression is the elastic potential energy: U s = ½ kx 2 The elastic potential energy can be thought of as the energy stored in the deformed spring.

Elastic Potential Energy, final The elastic potential energy stored in a spring is zero whenever the spring is not deformed (U = 0 when x = 0) 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 x 2 will always be positive

Example: An object moves in the xy-plane is subjected to a force that does 14 J of work on it as it moves from the origin to the position (2i - 3j) meters. Consider the three possible forces: F 1 = 4i - 2j ; F 2 = i - 4j ; F 3 = 2i -3j Which of the following statement is true? a) F 1 or F 2 could have done the work. b) Only F 1 could have done the work. c) F 1 or F 3 could have done the work. d) Only F 2 could have done the work. e) Only F 3 could have done the work.

Example:

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

Conservative Forces 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 A closed path is one in which the beginning and ending points are the same

Conservative Forces, cont Examples of conservative forces: Gravity Spring force We can associate a potential energy for a system with any conservative force acting between members of the system This can be done only for conservative forces In general: W C = -  U

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

Conservative Forces and Potential Energy 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  U is negative when F and x are in the same direction

Conservative Forces and Potential Energy The conservative force is related to the potential energy function through 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 Can be extended to three dimensions

Conservative Forces and Potential Energy – Check Look at the case of a deformed spring This is Hooke’s Law and confirms the equation for U U is an important function because a conservative force can be derived from it

Example: As a particle moves along the x axis it is acted by a conservative force. The potential energy is shown as a function of the coordinate x of the particle. Rank the labeled regions according to the magnitude of the force, least to greatest. 1) AB, CD, BC 2) AB, BC, CD 3) CD, BC, AB 4) BC, AB, CD

Potential Energy in Molecules There is potential energy associated with the force between two neutral atoms in a molecule which can be modeled by the Lennard-Jones function

Potential Energy Curve of a Molecule Find the minimum of the function (take the derivative and set it equal to 0) to find the separation for stable equilibrium The graph of the Lennard-Jones function shows the most likely separation between the atoms in the molecule (at minimum energy)

Energy Review Kinetic Energy Associated with movement of members of a system Potential Energy Determined by the configuration of the system Gravitational and Elastic Internal Energy Related to the temperature of the system

Types of Systems Nonisolated systems Energy can cross the system boundary in a variety of ways Total energy of the system changes Isolated systems Energy does not cross the boundary of the system Total energy of the system is constant

Ways to Transfer Energy Into or Out of A System Work – transfers by applying a force and causing a displacement of the point of application of the force Mechanical Waves – allow a disturbance to propagate through a medium Heat – is driven by a temperature difference between two regions in space

More Ways to Transfer Energy Into or Out of A System Matter Transfer – matter physically crosses the boundary of the system, carrying energy with it Electrical Transmission – transfer is by electric current Electromagnetic Radiation – energy is transferred by electromagnetic waves

Conservation of Energy Energy is conserved This means that energy cannot be created nor 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

Conservation of Energy, cont. Mathematically,  E system =  E system is the total energy of the system T is the energy transferred across the system boundary Established symbols: T work = W and T heat = Q Others just use subscripts

The Work-Kinetic Energy theorem (  K E = W ) is a special case of Conservation of Energy The full expansion of the above equation gives  K +  U +  E int = W + Q + T MW + T MT + T ET + T ER Conservation of Energy, cont.

Isolated System In any isolated system of objects that interact only through conservative forces, the total mechanical energy of the system remains constant. Remember: Total mechanical energy is the sum of the kinetic and potential energies in the system

Isolated System Conservation of Energy becomes: E system is all kinetic, potential, and internal energies This is the most general statement of the isolated system model If nonconservative (nc) forces are acting, some energy is transformed into internal energy  E system = 0 W nc =  E mech

Problem Solving Strategy – Conservation of Mechanical Energy for an Isolated System Categorize Define the system It may consist of more than one object and may or may not include springs or other sources of storing potential energy Determine if any energy transfers occur across the boundary of your system If there are transfers, use  E system =  T If there are no transfers, use  E system = 0

Problem Solving Strategy, cont Determine is there are any nonconservative forces acting If not, use the principle of conservation of mechanical energy

Problem-Solving Strategy, 2 Analyze Choose configurations to represent initial and final configuration of the system For each object that changes height, identify the zero configuration for gravitational potential energy For each object on a spring, the zero configuration for elastic potential energy is when the object is in equilibrium

Problem-Solving Strategy, 3 Analyze, cont Write expressions for total initial energy and total final energy Use and solve the appropriate conservation of Energy Equation

Example – Free Fall Determine the speed of the ball at y above the ground

Example – Free Fall, cont Analyze Apply Conservation of Energy K f + U gf = K i + U gi K i = 0, the ball is dropped Solving for v f Categorize System is isolated Only force is gravitational which is conservative

Example: A pendulum consists of a sphere of mass m attached to a light cord of length L. The sphere is released from rest at point A when the cord makes an angle  A with the vertical, and the pivot at point P is frictionless. Find the speed of the sphere when it is at the lowest point B.

Example: Two objects interact with each other and with no other objects. Initially object A has a speed of 5 m/s and object B has a speed of 10 m/s. In the course of their motion they return to their initial positions. Then A has a speed of 4 m/s and B has a speed of 7 m/s. We can conclude: A) the potential energy changed from the beginning to the end of the trip E) mechanical energy was increased by nonconservative forces D) mechanical energy was decreased by nonconservative forces C) mechanical energy was decreased by conservative forces B) mechanical energy was increased by conservative forces

Solution: Assuming that there are no friction. The mechanical energy of the ball is conserved: U = 0 h But, h = L – L cos(  A )

Solution cont’d: What is the tension T B in the string at point B. Newton’s Law: mg T

Example: A particle is released from rest at the point x = a and moves along the x axis subject to the potential energy function U(x) shown. The particle: A) moves to infinity at varying speed B) moves to x = b where it remains at rest C) moves to a point to the left of x = e, stops and remains at rest D) moves to a point to x = e, then moves to the left E) moves to x = e and then to x = d, where it remains at rest

Example: The launching mechanism of a toy gun consists of a spring of unknown spring constant. When the spring is compressed m, the gun, when fired vertically, is able to launch a 35.0 g projectile to a maximum height of 20.0 m above the position of the projectile before firing. Neglecting all resistive forces determine the spring constant k?

Solution: The mechanical energy of the Projectile-Spring-Earth is conserved:

Solution Cont’d: Find the speed of the projectile as it moves through the equilibrium position of the spring?

Example: Two block are connected by a light string that passes over a frictionless pulley, as shown. The block of mass m 1 lies on a horizontal table and is connected to a spring of constant k. The system is released from rest when the spring is unstretched. If the hanging block of mass m 2 falls a distance h before coming to rest, calculate the coefficient of kinetic friction between the table and the block.

Solution: W friction = E f – E i =  E mech W f = -f  h = -  k m 1 g  h E f = ½ kh 2 and E i = m 2 gh  -  k m 1 g  h = ½ kh 2 - m 2 gh The Change in Mechanical energy is equal to the work done by friction: System = m 1, m 2, earth and spring N.B. in this example if we include the table in our system:  E system = 0; i.e.  E mech +  E int = 0

Example – Spring Mass Collision Without friction, the energy continues to be transformed between kinetic and elastic potential energies and the total energy remains the same If friction is present, the energy decreases  E mech = -ƒ k d