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Mechanical Energy Essential Questions: Where does all this motion come from? What is the difference between dynamics and kinematics? Is energy real? What is work? Why does a gain in potential energy necessitate a loss in kinetic energy? How is energy conserved within systems? When can energy be described as not being conserved? 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 thought about 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, pressure, 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. Dynamics vs. Kinematics • The study of motion without regard to the underlying causes, particularly using time as a reference in a vector perspective is the study of kinematics. – This is what we have been doing all year thus far. • The study of motion via energy and work is considered one aspect of dynamics. – In may cases dynamics mirrors kinematics, but in some cases dynamics allows for much more simplistic and intuitive solutions to some VERY complex situations. – This aspect of our study will focus on one answer to the question "Why do objects move?“ and our answer to that question will relate to the physical quantity known as “work.” The Work/Energy Theorem Derivation: W = KE (= - PE) for later 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 abstract quantity that we study because certain properties of this “energy” variable help us figure stuff out about the universe. What is energy? • Simply put, energy is the potential to do work. • Work: The energy used/used up/consumed/transferred during a physical process. Conservative vs. Non-conservative forces Dynamics Conservative Forces Non-conservative Forces When work is done, energy is conserved within a closed system When work is done, energy is NOT conserved within a closed system (i.e., energy leaves the system) Reversible: Energy lost can be regained by reversing process ***Energy is still (always) conserved globally.*** Non-Reversible: Energy lost outside the system cannot be returned by reversing the process (2nd Law of Thermodyanimcs) FRICTION Force and Displacement: How do vectors play into this? • Energy is scalar—it doesn’t matter how the object gets there (in a conservative system), all that matters is if the state of the object’s energy before vs. after the event. • For work to be done by a force, that force must be applied in the direction of the displacement. Fx does work on the mower. Fx Fy Fy does not. In fact, Fy makes it harder to do work on the mower. Why? When force and displacement are in the same direction 𝑊 = 𝐹𝑥 When force and displacement are NOT in the same direction, but share a unidirectional component: 𝑊 = 𝐹𝑐𝑜𝑠𝜃 𝑥 𝑊 = 𝐹𝑥𝑐𝑜𝑠𝜃 When is work done? • When a force produces a displacement in the direction of that force. • When an object gains or loses kinetic or potential energy as a result of a force ebing applied. Can work be negative? • YES! This means only that work has been done against a force. • Often applied to objects being lifted “against” gravity: Applied force (lifting): W = Fx W = mgx Work done by gravity during lifting Wgrav = -mgx Negative work can imply that the flow of energy is opposite the direction of the force being referenced. Is work done? By what force? Did F create x? Did the energy of the object change? • A person lifts a book to place on a high bookshelf. • A person carries a sack of groceries at a constant height across a level floor. • A person carries a sack of groceries at a level height down a ramp. • A weight lifter holds a loaded barbell above his head. • An airplane comes in for a landing. • An airplane takes off. • A pitcher throws a baseball. Work, Energy, Force and Graphs Remember how we looked at the “area under” the velocity vs. time graph to calculate something? What did we calculate? Does above/below axis matter? How much work is done by this force as it displaces the object 0.5m? Does it matter if the axis went “under” the horizontal axis? What would that mean mathematically? What would that mean physically? Another example: What work is done by this force during the first 5m of displacement? During all 10m? Types of Energy: • Kinetic Energy – Anything moving has kinetic energy. • Potential Energy – Stored energy • For our purposes: PE’s are energies of position • In general: PE’s are energies of state • e.g., springs, chemical bonds, all matter, objects at heights, etc… Kinetic Energy (from the work energy theorem) 1 2 KE mv 2 Derivation of Gravitational PE Falling object 0 Gain and loss of energy Throwing a ball upwards Changes in energy are reversed from falling body. Falling: Gains KE, loses GPE Thrown up: Loses KE, gains GPE Conclusion: A gain in KE = loss in PE and A loss of KE = gain in PE +KE = -PE W = KE = -PE Complete work/energy theorem • The scenario in the prior problem assumes a closed, conservative system. Why is this premise only partly valid in real life (a.k.a. outside of magical physics land)? • Consider a bouncing ball. – Why? What does that have to do with anything? – Figure out where I’m going with this… 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 𝑊 ∆𝐸 𝑃= = 𝑡 𝑡 • 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. Know your context. AP-1: We will do these in a later unit. 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.