Work Done by Friction Review Conservation of Energy Power

Friction is a prime example of a non-conservative force. Let’s consider moving a book along a table. Looking down at the tabletop. A D1D1 μNμN B W friction, ABC = - μND 1 – μND 2  μND 3 = W friction, AC μNμN D3D3 Work by friction depends on path, so friction is a non- conservative force. C D2D2 μNμN

OSE U grav (y) = mgy (near surface of earth, y-axis up) You always get to choose where y=0, and where U grav =0. Choose it wisely to simplify your problem! Reminder: Spring Potential Energy An unstretched spring “wants” to stay unstretched. A stretched spring “wants” to go back to its equilibrium length. Reminder: Gravitational Potential Energy

According to Hooke’s law, the force that restores the spring to its equilibrium length is proportional to the amount of stretch:

OSE U spring (s) = ½ks 2 If s (x in the picture below) is defined as the difference between the equilibrium (unstretched) length of the spring and the stretched (or compressed length of the spring, then the potential energy of the spring is Let’s take s>0 to mean the spring is stretched, and s<0 to mean the spring is compressed (although because s is squared, the sign doesn’t “matter” for our results). Your text derives this equation on page 147.

Reminder: Conservation of Mechanical Energy We define the mechanical energy of a system to be OSE E = K + U. With this definition, we can write the work-energy principle like this: OSE E f – E i = [W other ] i  f.

If only conservative forces act on a system, the total mechanical energy is conserved: OSE if [W other ] i  f = 0, then E f – E i. This is the Law of Conservation of Mechanical Energy. It is a REALLY BIG IDEA. It’s a Hammer Equation!

If you have a great big nail to pound in, are you going to pound it with a dinky little screwdriver? Or a hammer?

E f – E i = [W other ] i  f K’s and U’s. Kinetic energies of any objects in our system. Also potential energies. Springs and Gravity! “What goes into E f and E i ?” “What kind of “things” do you know about that have potential energies?”

U spring and U grav ! For every spring and mass in the system. E f – E i = [W other ] i  f “OK, springs and gravity have potential energies. So what U’s go into the E of the above OSE?” No, not until next semester. “Anything else?”

Work done by any force that doesn’t come from a spring or gravity! E f – E i = [W other ] i  f “If springs and gravity go into E f and E i, what goes into W other ?” Work done by any force that doesn’t come from a spring or gravity! DUH! A look ahead: if we extend our definition of energy to include forms other than mechanical, we find that the total energy of a closed system is conserved. That’s a REALLY REALLY BIG IDEA. You’ll have to wait a bit for it.

I’ll work a simple example. A bowling ball of mass m is dropped on a spring of force constant K from a height of H above the spring. What is its speed after it has compressed the spring a distance d? Anybody try to care to solve this using kinematics? No, you don’t want to use kinematics. In fact, you can’t. You don’t know how to handle the non-constant spring force. But using energy methods makes this problem easy Problem Solving Using Conservation of Mechanical Energy

M m T N Ww T If two masses are connected by a massless, taut rope (so that the tension is the same everywhere along the rope), it is not necessary to include the work done by the tensions in W other. Special Dispensations in Energy Problems D D You do not need to justify this choice in your work, but always THINK before you make this choice! If mass M undergoes a displacement, mass m undergoes a displacement of the same magnitude. The two ’s are in “opposite” directions, relative to, so the net work is zero.

M m T N Ww T D Why can you do this? If your diagram shows it is valid, you do not need to justify this choice in your work, but always THINK before you make this choice! If your diagram specifically shows that the displacement of an object is perpendicular to the normal force on that object, you do not need to include the work done by in W other. Special Dispensations in Energy Problems

D F If your diagram specifically shows that the displacement of an object is perpendicular to a force acting on the object, you may immediately write in your solution, without further justification: WFWF 0 Special Dispensations in Energy Problems

D F If your diagram specifically shows that the displacement of an object is parallel to a force acting on the object, you may immediately write in your solution, without further justification: W F = FD Special Dispensations in Energy Problems

D F If your diagram specifically shows that the displacement of an object is antiparallel to a force acting on the object, you may immediately write in your solution, without further justification: W F = -FD Special Dispensations in Energy Problems

θ M D T If there is a tension force on a single object, or if that tension force is not perpendicular to the displacement, you may not “zero out” the work due to the tension force. Not a Special Dispensation in Energy Problems!

In a previous lecture, you learned that if only conservative forces act, the total mechanical energy of a system is conserved. If you include in your calculations the energy that exits or enters a system as a result of nonconservative forces, you find that the total energy of the system is conserved. “Total energy is neither increased nor decreased in any process.” 6.8 Other Forms of Energy: Energy Transformations and the Law of Conservation of Energy

Brief philosophical digression: conservation of total energy is “demanded” by precision experiments. This leaves you with a nagging feeling that maybe the experiments aren’t precise enough; maybe someday we’ll find that conservation of energy is not required. Most books are careful to point out that conservation of energy has been verified by every experiment done so far. It would be foolish to say that what we regard as truth 100 years from now will be the same as what we regard as truth today.

Just as Newtonian mechanics has been shown to be an approximation to quantum mechanics, valid only for “large” objects, theories of today are likely to be shown to be a subset of more encompassing theories. Conservation of energy is “bigger” than Newton’s laws, and I would expect to see conservation of energy still with us 100 years from now. However, the great conservation principles of physics, two of which we will study this semester, follow mathematically from symmetries in nature. Conservation of energy is a mathematical consequence of the symmetry of the universe under a translation in time.* *You get the same answer no matter when you set t=0.

The simple statement has many important consequences. See We will see the impact of this Law of Conservation of Energy a number of times this semester and next. Forces which reduce the total mechanical energy of a system are nonconservative. They are also called “dissipative” forces. “Total energy is neither increased nor decreased in any process...” 6.9 Energy Conservation with Dissipative Forces

If you include the energy generated in dissipative processes, you find that total energy in a closed system is conserved. (The Law of Conservation of Energy discussed in the previous section.) However, to be consistent with material shared with Physics 23, and to emphasize the importance of this idea, I’ll introduce two more OSE’s: In a sense, this information is already contained in our OSE E f – E i = [W other ] i  f. *OSE: E total,i = E total,f (for a closed system) OSE: E total = E +  *This is the BIG ONE.

E total = E +  E is mechanical energy. What kinds of energies are included in  ? “Thermal energy.” (Heat.) Acoustic energy (sound). Friction. (Heat.) Internal energy. All of these energies have something to do with the motion or configuration of atoms or molecules. Chemical energy.* Because chemical energy is not associated with a dissipative force, we could (if we wanted to) define a potential associated with chemical energy.

Next semester we will learn about electromagnetic energy. Our law of conservation of energy will still work. If you study special relativity (Einstein), you will find that the Law has to be restated as the Law of Conservation of Mass- Energy, but it still works. Examples: cannonball, lect10s.pdfcannonballlect10s.pdf Demonstration: trust in physics.

Power is defined as the rate at which work is done, or the rate at which energy is transformed (from one form to another): 6.10 Power Here is our textbook’s companion website. If you are going to take the MCAT, try some practice problems from chapters we have covered.website

The unit of power is 1 Joule/s = 1 Watt. If then Example: a 600 N marine climbs up a rope at a constant speed of 1.5 m/s. How much power does the force of gravity deliver to the marine’s body?

F g =mg F pull  The climber does positive work. But the angle is 0˚. v