PHY 151: Lecture System and Environments 6.2 Work Done by a Constant Force 6.3 Scalar Product of Two Vectors 6.4 Work Done by a Varying Force 6.5 Kinetic Energy and the Work-Energy Theorem 6.6 Potential Energy of a System 6.7 Conservative and Nonconservative Forces 6.8 Relationship between Conservative Forces and Potential Energy 6.9 Potential Energy for Gravitational and Electric Forces 6.10 Energy Diagrams and Equilibrium of a System

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

Energy Approach to Problems The energy approach to describing motion is particularly useful when the force is not constant A global approach to problems involving energy and energy transfers will be developed This could be extended to biological organisms, technological systems and engineering situations

PHY 151: Lecture 6 Energy of a System 6.1 System and Environment

Systems and Environments - 1 A system is a small portion of the Universe We will ignore the details of the rest of the Universe This is a simplification model A critical skill is to identify the system

Systems and Environments - 2 A valid system may: be a single object or particle be a collection of objects or particles be a region of space vary with time in size and shape

Systems and Environments - 3 There is a system boundary around the system The boundary is an imaginary surface It does not necessarily correspond to a physical boundary The boundary divides the system from the environment The environment is the rest of the Universe

PHY 151: Lecture 6 Energy of a System 6.2 Work Done by a Constant Force

Work Done by a Constant Force - 1 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  r of the displacement of the point of application of the force, and cos  where  is the angle between the force and the displacement vectors:

Work Done by a Constant Force - 2 The displacement is at the point of application of the force. A force does no work on the object if the force does not move through a displacement

Work Done by a Constant Force - 3 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. The normal force, n, and the gravitational force, mg, do no work on the object

Work Done by a Constant Force - 4 Work is a scalar quantity It can be either negative or positive –When an object is lifted, the work done by the applied force on the object is positive because the direction of that force is upward, in the same direction as the displacement –As an object is lifted, the work done by the gravitational force on the object is negative. The unit of work is a joule (J) 1 joule = 1 newton × 1 meter J = N · m

Example 6.1 A man cleaning a floor pulls a vacuum cleaner with a force of magnitude F = 50.0 N at an angle of 30.0  with the horizontal. Calculate the work done by the force on the vacuum cleaner as the vacuum cleaner is displaced 3.00 m to the right. Use the definition of work:

PHY 151: Lecture 6 Energy of a System 6.3 The Scalar Product of Two Vectors

The Scalar Product of Two Vectors - 1

The Scalar Product of Two Vectors - 2

The Scalar Product of Two Vectors - 3

The Scalar Product of Two Vectors - 4

Example 6.2

Evaluate the magnitudes of each vector using the Pythagorean theorem: Find the angle: Example 6.2

Use the equation for work: Example 6.3 (5.0i + 2.0j)N

PHY 151: Lecture 6 Energy of a System 6.4 Work Done by a Varying Force

Work Done by a Varying Force - 1 Consider a particle being displaced along the x axis under the action of a force that varies with position We cannot use W = F  r cos  because this relationship applies only when the force is constant in magnitude and direction

Work Done by a Varying Force - 2 Assume that during a very small displacement,  x, the force is constant –For that displacement, W 1  F  x –The total work is

Work Done by a Varying Force - 3 As displacement size approaches zero, we get The work done is equal to the area under the curve

Work Done by a Varying Force - 4 In general: –“ext” means work is done by an external agent

Example 6.4 A force acting on a particle varies with x. Calculate the work done by the force on the particle as it moves from x = 0 to x = 6.0 m. Evaluate the area of the rectangle: Evaluate the area of the triangle: Find the total work: 5

Work Done by a Varying Force - 5 Work done by a spring The force exerted by the spring is –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 (units: N/m) The force law for springs is called Hooke’s law –In vector form:

Work Done by a Varying Force - 6

Work Done by a Varying Force - 7 Force exerted by a spring is always directed opposite the displacement from equilibrium Because a spring force always acts towards the equilibrium position (x = 0), it is sometimes called a restoring force The work done by a spring is

Example 6.5 A spring is hung vertically, and an object of mass m is attached to its lower end. Under the action of the “load” mg, the spring stretches a distance d from its equilibrium position. (A) If a spring is stretched 2.0 cm by a suspended object having a mass of 0.55 kg, what is the force constant of the spring?

Example 6.5 –Apply the particle in equilibrium model to the object: –Apply Hooke’s law and solve for k: (B) How much work is done by the spring on the object as it stretches through this distance?

PHY 151: Lecture 6 Energy of a System 6.5 Kinetic Energy and the Work-Kinetic Energy Theorem

Kinetic Energy and the Work-Kinetic Energy Theorem - 1 Work is a mechanism for transferring energy into a system One possible result of doing work on a system is that the system changes its speed

Kinetic Energy and the Work-Kinetic Energy Theorem - 2 Kinetic energy is the energy of a particle due to its motion: K is the kinetic energy m is the mass of the particle v is the speed of the particle The work done by a net force on a particle equals the change in its kinetic energy:

inetic Energy and the Work-Kinetic Energy Theorem - 3 When work is done on a system and the only change in the system is in its speed, the net work done on the system equals the change in kinetic energy of the system

Example 6.6 A 6.0-kg block initially at rest is pulled to the right along a frictionless, horizontal surface by a constant horizontal force of 12 N. Find the block’s speed after it has moved 3.0 m. –Use the work-kinetic energy theorem:

Example 6.6 Solve for the final velocity: Substitute numerical values:

PHY 151: Lecture 6 Energy of a System 6.6 Potential Energy of a System

Potential Energy of a System - 1 The potential energy of a system is determined by the configuration of the system Consider a book lifted up from the surface of a table –When it is held above the table, it has “stored” energy due to its position –If released, the book will fall, converting this potential energy to the kinetic energy of motion

Potential Energy of a System - 2 For an object near the surface of the Earth: The work done by an external agent as object undergoes upward displacement is given by The gravitational potential energy is

Potential Energy of a System - 3 Gravitational potential energy depends only on the vertical height of the object above the surface of the Earth The important quantity is the difference in potential energy The difference is independent of the choice of reference configuration

Example 6.7 A trophy being shown off by a careless athlete slips from the athlete’s hands and drops on his toe. Choosing floor level as the y = 0 point of your coordinate system, estimate the change in gravitational potential energy of the trophy–Earth system as the trophy falls. Repeat the calculation, using the top of the athlete’s head as the origin of coordinates. mass = 2kg top of toe = 0.03 m trophy falls from height = 0.5 m

Example 6.7 Calculate the gravitational potential energy of the system just before trophy is released: Calculate the gravitational potential energy of the system when trophy reaches the athlete's toe: Evaluate the change in gravitational energy of the system:

Elastic potential energy Consider a block-spring system The work done by an external applied force is The elastic potential energy is defined as Elastic potential energy is the energy stored in the deformed spring Spring can be either stretched or compressed U s is always positive Potential Energy of a System - 4

Potential Energy of a System - 5

PHY 151: Lecture 6 Energy of a System 6.7 Conservative and Nonconservative Forces

Conservative and Nonconservative Forces - 1 Internal energy: the energy associated with the temperature of a system For example, when a book slides across a surface, friction does work and increases the internal energy of the surface Work is a transformation mechanism for energy

Conservative and Nonconservative Forces - 2 Conservative forces have two equivalent properties: 1. The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle 2. The work done by a conservative force on a particle moving through any closed path is zero. A closed path is one for which the beginning point and the endpoint are identical Nonconservative force: a force that does not satisfy properties 1 and 2 for conservative forces

Conservative and Nonconservative Forces - 3 Examples of conservative forces: Gravitational force Spring force In general, the work W int done by a conservative force on an object is given by U is potential energy

Conservative and Nonconservative Forces - 4 Mechanical energy: sum of the kinetic and potential energies of a system For conservative forces, the total mechanical energy of a system is constant A nonconservative forces cause a change in the mechanical energy of a system The frictional force is a nonconservative force

Demonstration Gravity is Conservative Force A 10-kg mass rises 20 m The 10-kg mass is then lowered 20 m to it’s starting position How much work is done by gravity?  Raise  W = mg(h i – h f ) = 10(9.8)(0 – 20) = J  Lower  W = mg(h i – h f ) = 10(9.8)(20 – 0) = J  Total Work = = 0 J

Work Done by Gravity Path Independence - 1 Mount Everest is 9000 m high How much work is done against gravity by a 70 kg person climbing straight up to the top of Mount Everest?  W = mg(h i – h f )=70(9.8)(-9000)=-6.17 x 10 6 N

Work Done by Gravity Path Independence - 2 How much work is done against gravity by a 70 kg person climbing a ramp to the top of Mt. Everest?  Because of the angle   Force along ramp = -mgsin   Distance moved, d, is along ramp  Angle between F and d is 0 degrees  W = Fd = Fdsin   h/d = sin   F = -mg(h/d)d = -mgh  Same work as climbing straight up Note: This shows that path doesn’t matter when gravity is the force

Demonstration Friction is Nonconservative Force A 10-kg mass slides 20 m on a table 10-kg mass then slides back 20 m to it’s starting position Coefficient of kinetic friction is 0.20 How much work is done by fricition?  Slide Forward  W = Fdcos  =  k (mg)dcos180  W = (0.2)(10)(9.8)(20)(-1) = -392 J  Slide Back  W = Fdcos  =  k (mg)dcos180  W = (0.2)(10)(9.8)(20)(-1) = -392 J  Total Work = = -784 J <> 0

PHY 151: Lecture 6 Energy of a System 6.8 Relationship between Conservative Forces and Potential Energy

Relationship Between Conservative Forces and Potential Energy - 1 For conservative forces, the work done is independent on the path Work only depends on initial and final coordinates For such a system, we can define a potential energy function U The work done within the system by the conservative force equals the negative of the change in the potential energy of the system The conservative force is related to the potential energy function through

PHY 151: Lecture 6 Energy of a System 6.9 Potential Energy for Gravitational and Electric Forces

Potential Energy for Gravitational and Electric Forces - 1 The gravitational force on a particle due to the Earth can be written in vector form as The gravitational force is conservative, so in general: U g is defined as zero when masses are infinitely far apart

Potential Energy for Gravitational and Electric Forces - 2 For many particles, the total gravitational potential energy of the system is the sum over all pairs of particles:

Example 6.8 A particle of mass m is displaced through a small vertical distance  y near the Earth’s surface. Show that in this situation the general expression for the change in gravitational potential energy reduces to the familiar relationship  U = mg  y.

Example 6.8 Combine the fractions in the general equation Evaluate r f – r i and r i r f if both initial and final positions of the particle are close to Earth's surface: Substitute:

PHY 151: Lecture 6 Energy of a System 6.10 Energy Diagrams and Equilibrium of a System

Energy Diagrams and Equilibrium of a System - 1 The motion of a system can often be understood qualitatively through a graph of its potential energy versus the position of a member of the system The force F s exerted by the spring on the block is related to U s through

Energy Diagrams and Equilibrium of a System - 2 When the block is placed at rest at the equilibrium position of the spring (x = 0), where F s = 0, it will remain there unless some external force F ext acts on it

Energy Diagrams and Equilibrium of a System - 3 If displaced, the block will accelerate toward x = 0 when released This means x = 0 is a position of stable equilibrium x = x max and x =  x max are called the turning points

Energy Diagrams and Equilibrium of a System - 4 Consider a particle moving under the influence of a conservative force shown in the figure If the particle is displaced from its equilibrium position, it will accelerate away from x = 0 The point x = 0 is a position of unstable equilibrium Neutral equilibrium occurs when U is constant over some region

Energy Diagrams and Equilibrium of a System - 5 There is potential energy associated with the force between two neutral atoms in a molecule which can be modeled by the Lennard- Jones function. 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)

PHY 151: Lecture 6 Energy of a System 6.11 Context Connection: Potential Energy in Fuels Skipped