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Mechanical Energy Where does all this motion come from? Where does all this motion come from? • Insofar as scientists understand our universe, there are essentially two “substances” that provide the make up of our existence: matter & energy • Thanks in great part to Albert Einstein, perhaps the wisest scientist the world has known to date, we know that matter and energy are the same thing! • It never ceases to amaze me how the seeming complexity of nature is in actuality very simple... E= 2 mc Energy Speed of light Mass (matter) Ever heard this? Bet you have, but I bet you never really knew what it meant... Let’s analyze the units of this quantity before we go on… So, back to the question: Where does all this motion come from? • Motion is just one manifestation of energy in our lives. • Other manifestations include heat, and electromagnetism, which at the fundamental level is actually caused by motion (vibration of charged particles). • Also, energy can be stored energy via chemical bonds, springs, gravitational differentials, and many other sources. So, energy is movement! Or, going back and looking at Newton’s Second Law: Forces cause accelerations which cause changes in motion. This becomes: Changes in energy result in forces which change/create motion. Note: This is glossing over a great deal of complicated physics, but any physicist will tell you this is the main idea. The Work/Energy Theorem Derivation: W = KE = - PE What does this mean? • Well, it sounds crazy, but energy is not really a real quantity. It is a derived product of other physical and chemical quantities. That does not mean that it is not real, it just means that it is an artificial quantity that we study because certain properties of this “energy” variable help us figure stuff out about the universe. Types of Energy: • Kinetic Energy – Anything moving has kinetic energy. • Potential Energy – Stored energy • e.g., springs, chemical bonds, all matter, objects at heights, etc… Kinetic Energy 1 2 KE mv 2 Potential Energy • Gravitational PE: – GPE is understood as the energy difference caused by the relative heights of different objects above the ground (at h=0m, GPE = 0 J) GPE mgh • Spring PE: – Spring PE is the energy stored in a compressed or stretched spring. It describes the spring’s tendency to snap back into its original shape. 1 2 Where k, is the spring constant or stiffness of SPE kx the spring and x is the displacement of the 2 spring from its neutral position. The Law of Conservation of Energy The total amount of energy in any system is constant. Energy can neither be created no destroyed, only changed to a new form. This principle is ALWAYS true. There are NO EXCEPTIONS. (just erroneous assumptions) Energybefore = Energyafter Work Work is the amount of energy used up in a mechanical process. W = Fx = KE = - PE Work-Energy Theorem Works nicely with the impulse-momentum theorem to help define a system. Impulse/Momentum is Newtonian Kinematics (VECTORS) Work/Energy is Dynamics (SCALAR) Power • Power is the RATE at which energy is used. It has many mathematical forms, but anytime you see any energy variable described over a period of time, you are dealing with power. • More power does not necessarily mean more energy… Units for Energy, etc. • The SI unit for energy is the Newton-Meter, also called the Joule (J). It is a (kg)(m2)/(s2). • Power is energy over time, so J/s which is also called a Watt (W) (like a 40 W light bulb…) – Be CAREFUL! Capital W is the unit for power, but as a variable is means work or energy. Using conservation of energy to help solve collision problems: • New definitions! – Elastic Collision: A collision where kinetic energy is conserved. – Inelastic Collision: A collision where kinetic energy is not conserved. – The TOTAL energy, sometimes called the “mechanical energy” in any system is conserved in any collision. Problem: • How do you solve an elastic collision system where both objects bounce off of one another elastically and both objects continue to move? • Example: – Two objects of mass m = 2.4 kg collide in an elastic collision. The first object is traveling 3.9 m/s to the right and the second is traveling 1.5 m/s also to the right. Find the final velocities of both objects after an elastic collision. How do you solve this? • Set up your conservation of momentum equation (m1v1i + m2v2i = …) – Notice that you have two unknown variables, v1f and v2f. • Now set up an equation that indicates conservation of kinetic energy. – Notice that you have the same two unknowns. • Solve the system of two equations either by substitution, or by elimination. REMEMBER! • You may only conserve kinetic energy in an elastic collision! • Certain systems in which objects collide inelastically may be solved using conservation of energy (all types of E) if more information is known about what happens either before or after the collision. You will not have to evaluate these systems in this course, but you will be responsible for using Cons. Of Energy to solve elastic collisions.