Potential Energy and Energy Conservation

1. Potential Energy and Energy Conservation

2. Warm-Up: The Flying (and Driving) Dutchman • Stuck in traffic? Can’t make to be in time in 9:00am Phys250 class? • What about the ability to fly in your own car? • Dutch design engineering firm has just developed a three-wheeled vehicle that travels both on ground and in air, via a set of unfolding helicopter blades. • The PALV (personal air and land vehicle), powered by a rotary engine, has a top ground speed of 125 mph (120 mph in the air) and can get between 60 and 70 miles per gallon of conventional gasoline. It can take off at close range, and can land vertically. • We will se how this project will be developed… • What is the thrust (forward force on the PALV) developed by the PALV Rotary engine with power output 213 hp when the vehicle is airborne and traveling in air horizontally at 120 mph? http://www.sparkdesign.nl

3. Warm-Up: Power Powerclimb • Runner with mass m runs up the stairs to the top of 443-m-tall Sears Tower. To lift herself there in 15 minutes (900 s), what must be her average power output in watts? Kilowatts? Horsepower? • Treat the runner as a particle of mass m. • Let’s find first how much work she must do against the gravity to lift herself at height h. • Another way: calculate average upward force and then multiply by upward velocity • Upward force here is vertical, average vertical component of velocity is (443m) / (900s) = 0.492 m/s

4. Gravitational Potential Energy

5. Gravitational Potential Energy

6. Gravitational Potential Energy • Energy associated with position is called potential energy • If elevation for which the gravitational potential energy is chosen to be zero has been selected then the expression for the gravitational potential energy as a function of position y is given by • Gravitational potential energyUgrav is associated with the work done by the gravitational force according to

7. ConservativewithNon-Conservative Forces

8. Conservative and Non-Conservative Forces • Work done by the conservative force only depends on the initial and finalpositions, and doesn’t depend on the path • Runner: gravitational force is conservative • From point 1 to point 2, same work • The work done by a conservative force has these properties: • It can always be expressed as the difference between the initial and final values of a potential energy function: DU = -W. • It is reversible. • It is independent of the path of the body and depends only on the starting and ending points. • When the initial and final points are the same (closed loop), the total work is zero. All forces which do not satisfy these properties are non-conservative forces.

9. Warm-Up: GravitationalPotential Energy • When this guy is in midair, only gravity does work on him (air resistance can be neglected) • Mechanical energy (sum of kinetic and gravitational potential energy) is conserved • E = K + U = const

10. Moving up K decreases U increases Moving down K increases U decreases W = m g Warm-Up: GravitationalPotential Energy • When this guy is in midair, only gravity does work on him (air resistance can be neglected) • Mechanical energy (sum of kinetic and gravitational potential energy) is conserved • E = K + U = const

11. Warm-Up: Work due toGravity Nearthe Earth Awayfrom the Earth

12. Warm-Up: Extinction 70 Million years ago Dinosaurs ruled the Earth They disappeared at the boundary between the Cretaceous and Tertiary periods (C-T boundary) Why ? Luis Alvarez (1911 – 1988) ~ Nobel Prize winner in Physics ~ suggested an asteroid impactmight be responsible

13. Warm-Up: Extinction Alvarez calculated the asteroid would need to be 10 km acrossand would leave a crater 150 km in diameter A crater off the Yucatan peninsula of Mexico has been identified as a possible impact site. Research on this crater has shown it is the result of a extra-terrestrial impact.

14. Warm-Up: Extinction Many asteroids and comets that cross the Earth’s path originate in the Oort cloud. This is a dense ring of asteroids that surrounds our solar system Most asteroids that hit the Earth originate in the inner Oort cloud that extends from 40 to 10,000 times the radius of the earth’s orbit from the sun.

15. Assume an asteroid started at rest in the middle of the inner Oort cloud (~5000 RE-S) Assume it is acted on primarily by the Sun Assume mass ~1016kg (10-km-rock) Warm-Up: Extinction Energy of the impact 1 Ton TNT = 4109 J Asteroid Impact:2x109 MT TNT Over 80,000 MPH !

16. 60º 30º 3 Quick Reminder: 30º-60º-90º Triangle B 2 1 C A

17. Elastic Potential Energy

18. Elastic Potential Energy • When you compress a spring: • If there is no friction, spring moves back • Kinetic energy has been “stored” in the elastic deformation of the spring • Rubber-band slingshot: the same principle • Work is done on the rubber band by the force that stretches it • That work is stored in the rubber band until you let it go • You let it go, the rubber gives kinetic energy to the projectile • Elastic body: if it returns to its original shape and size after being deformed

19. Elastic Potential Energy Spring is stretched It does negative work on block Equilibrium Spring relaxes It does positive work on block Spring is compressed Positive work on block Block moves from one position x1 to another position x2: how much work does the elastic (spring) force do on the block?

20. Elastic Potential Energy • Work done ON a spring to move one end from elongation x1 to a different elongation x2 • When we stretch the spring, we do positive workon the spring • When we relax the spring, work done on the spring is negative • Work done BY the spring • From N3L: quantities of work are negatives of each other • Thus, work Weldone by the spring • We can express the work done BY the spring in terms of a given quantity at the beginning and end of the displacement Elastic potential energy

21. Elastic Potential Energy • The graph of elastic potential energy for ideal spring is a parabola • For extension of spring, x>0 • For compression, x<0 • Elastic potential energy U is NEVER negative! • In terms of the change of potential energy:

22. Elastic Potential Energy • When a stretched spring is stretched greater, Wel is negative and U increases: greater amount of elastic potential energy is stored in the spring • When a stretched spring relaxes, x decreases, Wel is positive and U decreases: spring loses its elastic potential energy • More spring compressed OR stretched, greater its elastic potential energy

23. Elastic Potential Energy:Work - Energy Theorem • Work – Energy Theorem: Wtot=K2-K1, no matter what kind of forces are acting on the body. Thus: If only elastic force does work • Total mechanical energy E (the sum of elastic potential energy and kinetic energy) is conserved • Ideal spring is frictionless and massless • If spring has a mass, it also has kinetic energy • Your car has a mass of 1.2 ton or more • Suspension spring has a mass of few kg • So we can neglect spring’s mass in study of how the car bounces on its suspension

24. ElasticForce + other forces? • If forces other than elastic force also do work on the body, the total work is elastic force + other forces • The work done by all forces other than the elastic force equals the change in the total mechanical energy E of the system, where U is the elastic potential energy: • “System” is made up of the body of mass m and the spring of force constant k • When Wother is positive, E increases • When Wother is negative, E decreases

25. ElasticPotential Energy: Example • BothGravitational Potential Energy and Elastic Potential Energy • Spring with a body is hanged vertically • Bungee jumper • U1 and U2 then are initial and final values of the total potential energy • The work done by all forces other than the gravitational force or elastic force equals the change in the total mechanical energy E=K+U of the system, where U is the sum of the gravitational potential energy and the elastic potential energy

26. Force and Potential Energy

27. Force and Potential Energy • We have studied in detail two specific conservative forces, gravitational force and elastic force. • We have seen there is a definite relationship between a conservative force and the corresponding potential energy function. • The force on a mass in a uniform gravitational field is Fy = - mg. The corresponding potential energy function is U(y) = mgy. • The force exerted on a body by a spring of force constant k is Fx = - kx. The corresponding potential energy function is Us(x) = (1/2)kx2. • In some situations, you are given an expression for potential energy as a function of position and have to find corresponding force.

28. Force and Potential Energy Consider motion along a straight line, with coordinate x • Fx(x) is the x-component of force as function of x • U(x) is the potential energy as function of x • Work done by conservative force equals the negative of the change U in potential energy: • For infinitesimal displacement x, the work done by force Fx(x) during this displacement is ~ Fx(x)x (suppose that this interval is so small that the force will vary just a little) • In the limit x0: Force from potential energy, one dimension

29. Force and Potential Energy Force from potential energy, one dimension • In regions where U(x) changes most rapidly with x (i.e. where dU(x)/dx is large) the greatest amount of work is done during the displacement, and it corresponds to a large force magnitude • When Fx(x) is in positivex-direction, U(x) decreases with increasing x • Thus, Fx(x) and U(x) have opposite sign • Thus, the force is proportional to the negativeslope of the potential energy function • The physical meaning: conservative force always acts to push the system toward lower potential energy

30. Force and Potential Energy • Lets verify if this expression correctly gives the gravitational force and the elastic force when using the gravitational potential energy and the elastic potential energy: • The gravitational potential energy is linearly related to the elevation (i.e. constant slope) and the force is constant. • The elastic potential energy varies quadratically with position. The force varies in a linearly.

31. Force and Potential Energy

32. Force and Potential Energy in3D • Conservative force in three dimensions has components Fx, Fy, and Fz • Each component may be function of coordinates x, y, z • Potential energy change U is the function of coordinates as well • When particle moves a small distance x in x-direction, the force Fx is ~constant. It does NOT depend on Fyand Fz because these components of force are perpendicular to the displacement and do NO work Force from potential energy, three dimensions

33. Energy Diagrams

34. Energy Diagrams • In situations where a particle moves in one-dimension only under influence of a single conservative force it is very useful to study the graph of the potential energy as a function of positionU(x) • At any point on a graph of U(x), the force can be calculated as the negative of the slope of the potential energy function • Fx = - dU/dx • Example: Glider on an air track • Spring exerts a force Fx=-kx • Potential energy function U(x) • Limits of the motion are the points where U curve intersects the horizontal line representing the total mechanical energy E

35. Energy Diagrams • Any point where the force is zero is called equilibrium point • These are the "critical points" on the graph of U(x): • Points on the graph that are localminima correspond to "stable equilibria" since the force on particle tends to push it back toward the equilibrium point. • Points on the graph that are localmaxima correspond to "unstable equilibria" since force on particle tends to push it back toward the equilibrium point. • Points on the graph that are inflection points correspond to "neutral equilibria". • If the total mechanical energy is known, then the potential energy graph can be used to determine the speed at any point since E = K + U is constant (i.e. use K = E – U and then find speed)

36. Energy Diagrams

37. q Turning Points Bounds of the Motion y R x A Pendulum What is the motion? K can never be negative Motion is bounded