PHYS 1441 Section 501 Lecture 6 Monday June 21 2004 Dr Jaehoon Yu Non uniform circular motion Newton s Universal Law of Gravitation Work done by a constant force Kinetic Energy and Work Energy theorem Today s homework is homework 6 due 1pm next Wednesday Monday June 21 2004 PHYS 1441 501 Summer 2004 Dr Jaehoon Yu 1 Forces in Non uniform Circular Motion The object has both tangential and radial accelerations Fr F Ft What does this statement mean The object is moving under both tangential and radial forces F F r F t 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 How does the a ar2 at2 Monday June 21 2004 PHYS 1441 004 Spring 2004 2 acceleration look Dr Jaehoon Yu Example of Non Uniform Circular Motion A 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 with vertical T R m Fg mg What are the forces involved in this motion The gravitational force Fg and the radial force T providing tension tangential comp Radial comp F t mg sin mat at g sin v2 Fr T mg cos mar m R v2 T m g cos R At what angles the tension becomes maximum and minimum What are the tensions Monday June 21 2004 PHYS 1441 004 Spring 2004 Dr Jaehoon Yu 3 Motion in Resistive Forces Medium can exert resistive forces on an object moving through it due to viscosity or other types frictional property of the medium Some examples Air resistance viscous force of liquid etc These forces are exerted on moving objects in opposite direction of the movement 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 objects 2 Forces proportional to square of speed Large objects w reasonable speed Monday June 21 2004 PHYS 1441 004 Spring 2004 Dr Jaehoon Yu 4 Newton s Law of Universal Gravitation People have been very curious about the stars in the sky making observations 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 m1m2 m1m2proportional to the How would you write this Fg G 2 Fg 2 between With G square of the distance them principle mathematically r12 r12 G is the universal gravitational constant and its value is G 6 673 10 11 Unit N m 2 kg 2 This constant is not given by the theory but must be measured by experiment 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 2004 Dr Jaehoon Yu 5 More on Law of Universal Gravitation Consider two particles exerting gravitational forces to each other r 12 m1 m2 r F21 Two objects exert gravitational force on each other following Newton s 3rd law F12 Taking r 12 as the unit vector we can write the force m2 experiences as What do you think the negative sign mean F 12 m1m2 G 2 r 12 r It means that the force exerted on the particle 2 by particle 1 is attractive force pulling 2 toward 1 Gravitational force is a field force Forces act on object without physical contact between the objects at all times independent of medium between them How do you think the The gravitational force exerted by a finite size gravitational force on the spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass surface of the earth look of the distributions was concentrated at the center M Em Monday June 21 2004 PHYS 1441 004 Spring 2004 Dr Jaehoon Yu Fg G RE2 6 Example for Gravitation Using the fact that g 9 80m s2 at the Earth s surface find the average density of the Earth Since the gravitational acceleration is g So the mass of the Earth is Therefore the density of the Earth is G ME 11 M E 6 67 10 2 2 RE RE 2 R g ME E G 2 ME VE RE g 3g G 4 GRE 4 3 RE 3 9 80 3 3 5 50 10 kg m 4 6 67 10 11 6 37 106 Monday June 21 2004 PHYS 1441 004 Spring 2004 Dr Jaehoon Yu 7 Free Fall Acceleration Gravitational Force Weight 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 What would the gravitational acceleration be if the object is at an altitude h above the surface of the Earth mg g M Em RE2 ME G RE2 G M E m G M E m Fg mg G 2 2 R h r E ME g G RE h 2 What do these tell us about the gravitational acceleration 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 Monday June 21 2004 PHYS 1441 004 Spring 2004 Dr Jaehoon Yu 8 Example for Gravitational Force The 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 22x10 6N What is its weight when in its orbit The total weight of the station on the surface of the Earth is FGE mg ME M Em 6 G 2 4 22 10 N RE Since the orbit is at 350km above the surface of the Earth the gravitational force at that height is FO M Em RE2 mg G R h 2 R h 2 FGE E E Therefore the weight in the orbit is FO 2 RE2 6 37 106 6 6 F 4 22 10 3 80 10 N GE 2 2 6 5 RE h 6 37 10 3 50 10 Monday June 21 2004 PHYS 1441 004 Spring 2004 Dr Jaehoon Yu 9 Kepler s Laws Ellipse a b F1 c F2 Ellipses have two different axis major long and minor short axis and two focal points F1 F2 a is the length of a semi major axis b is the length of a semi minor axis Kepler lived in Germany and discovered the law s governing planets movement some 70 years before Newton by analyzing data 1 All planets move in elliptical orbits with the …
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