Conservative Forces Lecturer: Professor Stephen T. Thornton

Reading Quiz Is it possible for the gravitational potential energy of an object to be negative? A) yes B) no

Is it possible for the gravitational potential energy of an object to be negative? A) yes B) no Gravitational PE is mgh, where height h is measured relative to some arbitrary reference level where PE = 0. For example, a book on a table has positive PE if the zero reference level is chosen to be the floor. However, if the ceiling is the zero level, then the book has negative PE on the table. It is only differences (or changes) in PE that have any physical meaning.

Discussed kinetic energy Work-energy theorem Last Time Discussed kinetic energy Work-energy theorem

Today Conservative and nonconservative forces Gravitational potential energy Other kinds of potential energy Conservation of mechanical energy

Conservation of Energy A conservative force does zero total work on any closed path. The work done by a conservative force in going from an arbitrary point A to an arbitrary point B is independent of the path from A to B.

Doing Work Against Gravity Energy is reclaimed in this case.

Doing Work Against Gravity Work done by gravity = -mgh d d mg Energy is reclaimed in this case.

Work Done by Gravity on a Closed Path is Zero. Use box to show this.

Work Done by Friction on a Closed Path is Not Zero. Floor (top view) Use box as demo.

Conservative Forces Gravity Springs Nonconservative Forces Friction Tension

Potential Energy When we do work, say to lift a box off the floor, then we give the box energy. We call that energy potential energy. Potential energy, in a sense, has potential to do work. It is like stored energy. However, it only works for conservative forces.

Do potential energy demo. Burn string and let large mass drop.

Notes on potential energy Potential energy is part of the work-energy theorem. Potential energy can be changed into kinetic energy. Think about gravity for a good example to use. There is no single “equation” to use for potential energy. Remember that it is only useful for conservative forces.

Potential Energy In raising a mass m to a height h, the work done by the external force is . We therefore define the gravitational potential energy at a height y above some reference point: Figure 8-4. Caption: A person exerts an upward force Fext = mg to lift a brick from y1 to y2 . .

Definition of potential energy We will (sometimes) use a subscript on Wc to remind us about conservative forces. This doesn’t work for friction. SI unit is the joule (still energy).

Remember gravity The work done by a conservative force is equal to the negative of the change in potential energy. Hold a box up. It has potential energy. Drop the box. Gravity does positive work on the box. The change in the gravitational potential energy is negative. The box has less potential energy when it is on the floor. Use the box in this slide.

More potential energy (PE) notes Gravitational potential energy = mgh Only change in potential energy U is important. There is no absolute value of PE. We choose the zero of PE to be at the most convenient position to solve problem.

Gravitational potential energy Because we can choose the “zero” of potential energy anywhere we want, it might be convenient to place it at y = 0 (but not always!).

Where might we choose the zero of potential energy to be here?

Do demos Loop the loop Bowling ball (wrecking ball video) Hopper popper http://www.youtube.com/watch?v=Rx28g0aqfIk

is B.

An object can have potential energy by virtue of its surroundings. Familiar examples of potential energy: A compressed (or wound-up) spring A stretched elastic band An object at some height above the ground

Potential Energy General definition of gravitational potential energy: For any conservative force:

In one dimension, We can invert this equation to find F(x) if we know U(x): In three dimensions:

Gravitational Potential Energy Boy does +mgy work to climb up to y. (Gravity does negative work, -mgy). He has potential energy mgy. Gravity does work on boy to bring him down. The potential energy is converted into kinetic energy. I like this slide.

Conservation of mechanical energy Mechanical energy E is defined to be the sum of K + U. Mechanical energy is conserved. Only happens for conservative forces.

Conservation of Mechanical Energy In the image on the left, the total mechanical energy at any point is: Figure 8-7. Caption: The rock’s potential energy changes to kinetic energy as it falls. Note bar graphs representing potential energy U and kinetic energy K for the three different positions.

Solving a Kinematics Problem Using Conservation of Energy E = mgh E = 0

High Jump. In the high jump, the kinetic energy of an athlete is transformed into gravitational potential energy without the aid of a pole. With what minimum speed must the athlete leave the ground in order to lift his center of mass 2.10 m and cross the bar with a speed of 0.70 m/s? Giancoli, 4th ed, Problem 8-13

Conceptual Quiz You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PEi + KEi. You watch the leaf float down until just before it hits the ground, at which point it has final total energy PEf + KEf. How do these total energies compare? A) PEi + KEi > PEf + KEf B) PEi + KEi = PEf + KEf C) PEi + KEi < PEf + KEf D) impossible to tell from the information provided Click to add notes

Conceptual Quiz You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PEi + KEi. You watch the leaf float down until just before it hits the ground, at which point it has final total energy PEf + KEf. How do these total energies compare? A) PEi + KEi > PEf + KEf B) PEi + KEi = PEf + KEf C) PEi + KEi < PEf + KEf D) impossible to tell from the information provided As the leaf falls, air resistance exerts a force on it opposite to its direction of motion. This force does negative work, which prevents the leaf from accelerating. This frictional force is a nonconservative force, so the leaf loses energy as it falls, and its final total energy is less than its initial total energy. Follow-up: What happens to leaf’s KE as it falls? What net work is done?