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# TTU PHYS 1408 - Conservation of Energy

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Physics 1408-002 Principles of Physics Sung-Won Lee [email protected] Lecture 14 – Chapter 8 & 9 – February 25, 2009 Announcement I Lecture note is on the web Handout (6 slides/page) http://highenergy.phys.ttu.edu/~slee/1408/ *** Class attendance is strongly encouraged and will be taken randomly. Also it will be used for extra credits. HW Assignment #6 will be placed on MateringPHYSICS today, and is due by 11:59pm on Wendseday, 3/4 Announcement II SI session by Reginald Tuvilla Monday 4:30 - 6:00pm - Holden Hall 106 Thursday 4:00 - 5:30pm - Holden Hall 106 SI sessions will be at the following times and location. Chapter 8 Conservation of Energy •! Conservative and Non-conservative Forces •! Potential Energy •! Mechanical Energy and Its Conservation •! Problem Solving Using Conservation of Mechanical Energy •! The Law of Conservation of Energy •! Energy Conservation with Dissipative Forces: Solving Problems •! Gravitational Potential Energy and Escape Velocity •! Power 8-7 Gravitational Potential Energy and Escape Velocity Far from the surface of the Earth, the force of gravity is not constant: The work done on an object moving in the Earth’s gravitational field is given by: Arbitrary path of particle of mass m moving from point 1 to point 2. 8-7 Gravitational P.E. and Escape Velocity Solving the integral gives: We can define gravitational potential energy: Gravitational potential energy plotted as a function of r, the distance from Earth’s center. Valid only for points r > rE , the radius of the Earth.8-7 Gravitational Potential Energy and Escape Velocity Example: Package dropped from high-speed rocket. A box of empty film canisters is allowed to fall from a rocket traveling outward from Earth at a speed of 1800 m/s when 1600 km above the Earth’s surface. The package eventually falls to the Earth. Estimate its speed just before impact. Ignore air resistance. 8-7 Gravitational Potential Energy and Escape Velocity If an object’s initial kinetic energy is equal to the potential energy at the Earth’s surface, its total energy will be zero. The velocity at which this is true is called the escape velocity; for Earth: 8.8 Power •! The time rate of energy transfer •! The average power is given by Instantaneous Power •! The instantaneous power is the limiting value of the average power as !t approaches zero •! This can also be written as Power Generalized •! Power can be related to any type of energy transfer •! In general, power can be expressed as Units of Power •! The SI unit of power is called the [watt] –! 1 watt = 1 joule / second = 1 kg . m2 / s2 •! US Customary system is horsepower: 1 hp = 746 W •! Unit of energy can be defined in terms of units of power. –! 1 kWh (kilowatt-hour)= (1000 W)(3600 s) = 3.6 x106 J Summary of Chapter 8 •! Gravitational potential energy: Ugrav = mgy. •! Elastic potential energy: Uel = ! kx2. •! For any conservative force: •! Total mechanical energy is the sum of kinetic and potential energies. •! Additional types of energy are involved when nonconservative forces act. •! Gravitational potential energy: •! Power: Chapter 9 Linear Momentum •! Conservative Momentum and Its Relation to Force •! Conservation of Momentum •! Collisions and Impulse •! Conservation of Energy and Momentum in Collisions •! Inelastic Collisions •! Collisions in 1-, 2- or 3-Dimensions •! Center of Mass (CM) and Translational Motion •! Systems of Variable Mass; Rocket PropulsionMomentum and Impulse Interaction forces between systems can be very complex. Example: a tennis ball colliding with a racquet Our goal is to find a relationship between the velocities of the objects before and after the interaction. Profile of the force during a collision. Microscopic view of a “bounce”. A collision is a short-duration interaction b/w 2 objects. Typical collision times are b/w 1 and 10 ms, depending on the materials involved. The harder the objects, the shorter the contact time. During contact the materials compress and a large spring-like force is exerted, called an impulsive force, which repels the object back apart. Let’s write F(t) for the force since it changes with time 9-1 Momentum and Its Relation to Force Momentum is a vector symbolized by the symbol , and is defined as The rate of change of momentum is equal to the net force: This can be shown using Newton’s second law. "momentuDefinition: m" p mv! =! !; ; ;x x y y z zp mv p mv p mv= = =nd( ) 2 Law:dv d mv dpF ma mdt dt dt= = = =! ! !!!212 1( )tx x x xtp p p F t dt! = " =#Momentum is a vector quantity Force changes momentum Momentum The product of a particle’s mass and velocity is called the momentum In terms of components: Newton formulated his 2nd law in terms of momentum: 9-2 Conservation of Momentum During a collision, measurements show that the total momentum does not change: Momentum is conserved in a collision of two balls, labeled A and B. 9-2 Conservation of Momentum Conservation of momentum can also be derived from Newton’s laws. A collision takes a short enough time that we can ignore external forces. Since the internal forces are equal and opposite, the total momentum is constant. Collision of two objects. Their momenta before collision are pA and pB, and after collision are pA’ and pB’. At any moment during the collision each exerts a force on the other of equal magnitude but opposite direction.Conservation of Momentum OK, we’ll now derive a law which relates the initial and final momenta of interacting objects. Consider 2 particles, with velocities directed along x-axis, which collide and then bounce apart. (vix)1 and (vix)2 i: initial, x=direction, 1=object1, 2=object2 Conservation of Momentum 12 on 121 on 2 2 on 1( )( )( )( ) ( )xxxx xd pFdtd pF Fdt== = !1 2 1 22 on 1 2 on 1[( ) ( ) ] ( ) ( )( ) ( ) 0x x x xx xd p p d p d pdt dt dtF F+= += ! =1 2( ) ( ) constantx xp p+ =1 2 Before 1 2 After[( ) ( ) ] =[( ) ( ) ]x x x xp p p p+ +Newton’s 2nd law for each particle during the collision The Law of Conservation of Momentum: The sum of the momenta after the collision equals the sum of the momenta before the collision!! 9-2 Conservation of Momentum For more than two objects, •! Conservation of momentum can be expressed mathematically in various ways ptotal = p1 + p2 = constant p1i + p2i= p1f + p2f 9-2 Conservation of Momentum Momentum conservation works for a rocket as

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