Physics 1220 Week 8 Notes
Physics 1220 Week 8 Notes PHYS1220
Popular in Physics with Calculus I
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This 8 page Class Notes was uploaded by Jennifer Asselin on Thursday September 29, 2016. The Class Notes belongs to PHYS1220 at Clemson University taught by Pooja Puneet in Summer 2016. Since its upload, it has received 5 views. For similar materials see Physics with Calculus I in PHYSICS (PHY) at Clemson University.
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Date Created: 09/29/16
Physics 1220 Chapter 8 Outline • Two approaches to studying mechanics o The Force Approach: uses forces and Newton’s Laws of Motion o The Conservation Approach: based on the idea that certain quantities remain unchanged during a process or activity • Energy o Describes the state of a particle, object or system o Not a material substance o Scalar quantity that describes the state (motion and configuration) of a system • Kinetic Energy (K) describes the motion of a system o One particle: ! = !!! ! ! o More than one particle: !!! ! ! ! o The unit of energy is Joules: 1J = 1N*m • Potential Energy (U) o Describes a system that has no motion, but that may still have energy associated with the arrangement of the particles in it o Only used to describe a system consisting of two or more particles that interact with each other via one or more internal forces o Can only be associated with internal conservative forces o Cannot be used to describe a single particle; two particles must be in relation o ! = !"#, where y is the displacement of height !! o ∆! = ! − ! ! − ! ! !!!" ! o Units: 1Nm = 1J • To make calculations easier, we usually choose a particular configuration to be the reference configuration and assign it a potential energy of zero • Gravitational Potential Energy: ! !!= !"# o Potential Energy is a scalar, so positive and negative signs indicate relative change, not direction like a vector • Only a vertical change in a particle’s position results in a change in the potential energy of the Earth-particle system • The change in potential energy of a system is path independent o It doesn’t matter if you go straight there or run around in circles on the way to the final position o The displacement determines a change in potential energy only o ∆! = !"( ! − ! ! ! ! ! ! • Gravitational Potential Energy: ! !!= −! ! ! ! ! • In any problem, you may have to use ! ! =!!"#, or ! ! = −! ! ! , but you will not use both for the same reference system • Use ! ! when working with a system containing a large object (like a planet and a small object near the surface of the small one • Elastic potential energy: energy stored by a spring o ! =! !!! ! ! o ∆! = !!!− !! ! ! ! • ! is interchangeable with other axes depending on the reference system • We usually place the origin of the coordinate system at the relaxed position • Potential energy can only be associated with two or more particles in one system, so the conservation approach requires that we choose a system of two or more particles • Isolated System o Does not interact exert force on the environment, and the environment does not exert force on it o Idealization, not really possible (like the particle system) • Mechanical Energy: ! = ! + ! • The principle of conservation of mechanical energy: if only conservative forces are acting, the mechanical energy of an isolated system is conserved • Equations representing the conservation of mechanical energy: o ∆! = 0 o ! +!! = ! + !! ! o ∆! + ∆! = 0 • Bar charts (Bar Graphs): help visualize the conservation of mechanical energy o The sum of the K (kinetic energy) bar and the U (potential energy) bar must always equal the E (mechanical energy) bar o K and E might change, but the sum is always the same o There are three bars: U, K, and E § U+K=E § E will stay the same over time • Examples of conservative forces: o Hooke’s Law spring force § Potential Energy: !! o Gravity § Potential Energy: !!!" ! ! • We can add these together to find total potential energy: ! = ! ! ! ! • Mechanical Energy is then: ! = ! + ! +!! ! • Potential Energy Curves: graph potential energy versus relative position • Energy Graph: represents mechanical energy (E), potential energy (U), and kinetic energy (K), on a single set of axes o Contains a U (potential energy) curve and a horizontal line representing mechanical energy o Kinetic energy is found by subtracting the function values on the y axis: K = E – U o Kinetic energy cannot be negative o Cannot be in a configuration where U>E because potential energy cannot be greater than the total mechanical energy § This is known as a forbidden configuration in the system • Force Approach o Involves vectors, like force, acceleration, velocity, displacement o Free body diagrams o Newton’s laws of motion • Conservation of Mechanical Energy Approach: o Involves scalars: kinetic and potential energy, speed, and components of position only o Bar charts o Energy graphs • Mechanical energy of a small mass m orbiting a large mass (like a planet) M: ! = − (! !"# ) ! ! • Consider this two-particle, isolated system with conserved energy. One particle moves in an elliptical orbit, and another moves in a circular orbit around the sun o The two orbits intersect at points I and J !"# o Potential Energy at I and J: !!= − ! o Kinetic Energy at I and J: ! = ! !"# = !!! ! ! ! ! ! !" o So, the Comet’s speed at I or J is:!! = ! o The speed of the comet in circular orbit is constant, but the comet in elliptical orbit varies o The diagram below shows the two orbiting particles described above. I and J and the locations where the solid and dashed lines cross: Image from: http://hildaandtrojanasteroids.net/TA033ellipticalspeed.jpg
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