PHYS 1443 – Section 501Lecture #19Lec. 11: Newton+KeplerKepler’s Laws & EllipseExample of Kepler’s Third LawLec. 13: Potential EnergyLec: 14 conservative+non-cons forcesWork Done by non-conservative ForcesLec. 15: Grav. field, escape speed, powerLec.16: Momentum,Impulse,collisionsExample for CollisionsLec. 17: Center of MassExample for Center of MassMotion of a Group of ParticlesPHYS 1443 – Section 501Lecture #19Monday Apr 5, 2004Dr. Andrew Brandt•Review for test weds Apr. 7Monday, April 5, 2004 1PHYS 1443-501, Spring 2004Dr. Andrew BrandtLec. 11: Newton+Kepler1110673.6−×=G21221rmmGFg=22/ kgmN ⋅1222112ˆrrmmGF −=g2EERMG==9.8 m/s2Monday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt2Kepler’s Laws & EllipseF1F2bcaEllipses have two different axis, major (long) and minor (short) axis, and two focal points, F1& F2a is the length of a semi-major axisb is the length of a semi-minor axis1. All planets move in elliptical orbits with the Sun at one focal point.2. The radius vector drawn from the Sun to a planet sweeps out equal area in equal time intervals. (Angular momentum conservation)3. The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit.Monday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt3Example of Kepler’s Third LawCalculate the mass of the Sun using the fact that the period of the Earth’s orbit around the Sun is 3.16x107s, and its distance from the Sun is 1.496x1011m.324rGMs⎟⎟⎠⎞⎜⎜⎝⎛=π3rKs=Using Kepler’s third law.2T2324rGTπ⎛⎞=⎜⎟⎝⎠The mass of the Sun, Ms, issM()231111 7 241.496 106.67 10 (3.16 10 )π−⎛⎞=××⎜⎟×× ×⎝⎠kg301099.1 ×=Monday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt4A man cleaning a floor pulls a vacuum cleaner with a force of magnitude F=50.0N at an angle of 30.0owith East. Calculate the work done by the force on the vacuum cleaner as it is displaced by 3.00m to East.Lec. 12: Work and KE()=⋅∑dF()θcosdF∑MF30οMd=WJW 13030cos00.30.50 =××=ο()∫∑=fixxixdxFnetW )(variable force()max max20012xxsW F dx kx dx kx==−=∫∫spring forceKEKEKEmvmvWifif∆=−=−=222121work and KEMonday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt5Lec. 13: Potential EnergyEnergy associated with a system of objects Î Stored energy which has Potential or possibility to work or to convert to kinetic energyffiiMPEKEPEKEE+=+≡cons of energymgyUg≡grav PE221kxUs≡elastic PEMonday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt6Lec: 14 conservative+non-cons forcesmghA ball of mass m is attached to a light cord of length L, making up a pendulum. The ball is released from rest when the cord makes an angle θAwith the vertical, and the pivoting point P is frictionless. Find the speed of the ball when it is at the lowest point, B.=hUsing the principle of mechanical energy conservationffiiUKUK+=+Compute the potential energy at the maximum height, h. Remember where 0 is.mgmmθALTh{()AALLLθθcos1cos−=−mghUi=()AmgLθcos1−=PE0KE0mv2/2=+mgh0()=−AmgLθcos1221mv()AgLvθcos122−=()AgLvθcos12 −=∴BMonday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt7Work Done by non-conservative ForcesMechanical energy of a system is not conserved when any one of the forces in the system is a non-conservative force.Two kinds of non-conservative forces:Applied forces: Forces that are external to the system. These forces can take away or add energy to the system. So the mechanical energy of the system is no longer conserved.If you were to carry around a ball, the force you apply to the ball is external to the system of ball and the Earth. Therefore, you add kinetic energy to the ball-Earth system. ;KWWgyou∆=+UWg∆−=UKWWappyou∆+∆==Kinetic Friction: Internal non-conservative force that causes irreversible transformation of energy. The friction force causes the kinetic and potential energy to transfer to internal energydfKWkfrictionfriction−=∆=dfUKEEEkif−=∆+∆=−=∆Monday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt8Lec. 15: Grav. field, escape speed, powergmFg=rRGMEEˆ2−=U∆ifUU −=()∫−=firrdrrFrmGmU21−=Average power tWP∆∆===∆∆≡→∆dtdWtWPt 0limInstantaneous power ()=⋅ sdtdF=⋅ vFθcosFvMonday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt9Lec.16: Momentum,Impulse,collisionsA new concept of linear momentum can also be used to solve physical problems, especially the problems involving collisions of objects.vmp =Linear momentum of an object whose mass is m and is moving at a velocity of v is defined as 1. Momentum is a vector quantity.2. The heavier the object the higher the momentum3. The higher the velocity the higher the momentum4. Its unit is kg.m/s Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant.pdtFIfitt∆=≡∫Monday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt10Monday, April 5, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt11Example for CollisionsA car of mass 1800kg stopped at a traffic light is rear-ended by a 900kg car, and the two become entangled. If the lighter car was moving at 20.0m/s before the collision what is the velocity of the entangled cars after the collision?ipThe momenta before and after the collision areWhat can we learn from these equations on the direction and magnitude of the velocity before and after the collision?m120.0m/sm2vfm1m2Since momentum of the system must be conservedfipp =The cars are moving in the same direction as the lighter car’s original direction to conserve momentum. The magnitude is inversely proportional to its own mass.Before collisionAfter collisioniivmvm2211+=ivm220 +=fpffvmvm2211+=()fvmm21+=()fvmm21+ivm22=fv()2122mmvmi+=smii/ 67.618009000.20900=+×=Lec. 17: Center of MassWe’ve been solving physical problems treating objects as sizelesspoints with masses, but in realistic situation objects have shapes with masses distributed throughout the body. Center of mass of a system is the average position of the system’s mass and represents the motion of the system as if all the mass is on the point. Consider a massless rod with two balls attached at either end.212211mmxmxmxCM++≡The total external force exerted on the system of total mass M causes the center of mass to move at an acceleration given by as if all the mass of the system is concentrated on
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