Monday, June 21, 2004 PHYS 1441-501, Summer 2004Dr. Jaehoon Yu1PHYS 1441 – Section 501Lecture #6Monday, June 21, 2004Dr. Jaehoon Yu• Non-uniform circular motion• Newton’s Universal Law of Gravitation• Work done by a constant force• Kinetic Energy and Work-Energy theoremToday’s homework is homework #6, due 1pm, next Wednesday!!Monday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu2Forces in Non-uniform Circular MotionThe object has both tangential and radial accelerations.What does this statement mean?The object is moving under both tangential and radial forces.FrFtFtrFFF +=These forces cause not only the velocity but also the speed of the ball to change. The object undergoes a curved motion under the absence of constraints, such as a string. 22traaa +=How does the acceleration look?Monday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu3Example of Non-Uniform Circular MotionA ball of mass m is attached to the end of a cord of length R. The ball is moving in a vertical circle. Determine the tension of the cord at any instant when the speed of the ball is v and the cord makes an angle θ with vertical. TmWhat are the forces involved in this motion?=∑tFThe gravitational force Fgand the radial force, T, providing tension. θRFg=mgAt what angles the tension becomes maximum and minimum. What are the tensions?θsingat==∑rF⎟⎟⎠⎞⎜⎜⎝⎛−=θcos2gRvmTtangential comp.Radial comp.=θsinmgtma=+θcosmgT=rmaRvm2Monday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu4Motion in Resistive ForcesMedium can exert resistive forces on an object moving through it due to viscosity or other types frictional property of the medium.These forces are exerted on moving objects in opposite direction of the movement. Some examples? These forces are proportional to such factors as speed. They almost always increase with increasing speed. Two different cases of proportionality: 1. Forces linearly proportional to speed: Slowly moving or very small objects2. Forces proportional to square of speed: Large objects w/ reasonable speedAir resistance, viscous force of liquid, etcMonday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu5Newton’s Law of Universal GravitationPeople have been very curious about the stars in the sky, makingobservations for a long time. But the data people collected have not been explained until Newton has discovered the law of gravitation. Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.How would you write this principle mathematically?21221rmmFg∝1110673.6−×=GG is the universal gravitational constant, and its value isThis constant is not given by the theory but must be measured by experiment.With G21221rmmGFg=Unit?22/ kgmN ⋅This form of forces is known as an inverse-square law, because the magnitude of the force is inversely proportional to the square of the distances between the objects.Monday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu6It means that the force exerted on the particle 2 by particle 1 is attractive force, pulling #2 toward #1.More on Law of Universal GravitationConsider two particles exerting gravitational forces to each other.Gravitational force is a field force: Forces act on object without physical contact between the objects at all times, independent of medium between them.1222112ˆrrmmGF −=The gravitational force exerted by a finite size, spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass of the distributions was concentrated at the center.m1m2rF21F1212ˆrTwo objects exert gravitational force on each other following Newton’s 3rdlaw.Taking as the unit vector, we can write the force m2experiences as12ˆrWhat do you think the negative sign mean?gFHow do you think the gravitational force on the surface of the earth look?2EERmMG=Monday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu7Example for GravitationUsing the fact that g=9.80m/s2at the Earth’s surface, find the average density of the Earth.gSince the gravitational acceleration is So the mass of the Earth is GgRMEE2=Therefore the density of the Earth is ρ2EERMG=2111067.6EERM−×=EEVM=324EERGgR3=πEGRgπ43=33611/1050.51037.61067.6480.93mkg×=×××××=−πMonday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu8Free Fall Acceleration & Gravitational ForceWeight of an object with mass m is mg. Using the force exerting on a particle of mass m on the surface of the Earth, one can get•The gravitational acceleration is independent of the mass of the object•The gravitational acceleration decreases as the altitude increases•If the distance from the surface of the Earth gets infinitely large, the weight of the object approaches 0.What would the gravitational acceleration be if the object is at an altitude h above the surface of the Earth?mgWhat do these tell us about the gravitational acceleration?gF2EERmMG=g2EERMG='mg=2rmMGE=()2hRmMGEE+='g()2hRMGEE+=Monday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu9Example for Gravitational ForceThe international space station is designed to operate at an altitude of 350km. When completed, it will have a weight (measured on the surface of the Earth) of 4.22x106N. What is its weight when in its orbit?The total weight of the station on the surface of the Earth isTherefore the weight in the orbit isGEFOFSince the orbit is at 350km above the surface of the Earth, the gravitational force at that height isMEEEOFmg=2EERmMG=N61022.4 ×='mg=()2hRmMGEE+=()GEEEFhRR22+=()GEEEFhRR22+=()()N66256261080.31022.41050.31037.61037.6×=×××+××=Monday, June 21, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu10Kepler’s Laws & EllipseKepler lived in Germany and discovered the law’s governing planets’ movement some 70 years before Newton, by analyzing data.Newton’s laws explain the cause of the above laws. Kepler’s third law is the direct consequence of law of gravitation being inverse square law.1. 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.F1F2bcaEllipses have two
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