Physics 121 Thursday February 28 2008 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Thursday February 28 2008 Course Information Quiz Topics to be discussed today The center of mass Conservation of linear momentum Systems of variable mass Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Thursday February 28 2008 Homework set 5 is now available on the WEB and will be due next week on Saturday morning March 8 at 8 30 am This homework set contains WeBWorK problems and a video analysis The most effective way to work on the assignment is to tackle 1 or 2 problems a day If you run into problems please contact me and I will try to help you solve your problems Physics 121 problems that is Frank L H Wolfs Department of Physics and Astronomy University of Rochester 1 Homework Set 5 WeBWorK problems Finding U x The escape velocity Rocket motion Conservation of energy including friction losses Collision force Inelastic collisions and conservation of energy Frank L H Wolfs 2D collision Department of Physics and Astronomy University of Rochester Homework Set 5 WeBWorK problems Elastic collisions and conservation of energy Frank L H Wolfs 2D collision Explosion in 1D Department of Physics and Astronomy University of Rochester Physics 121 Thursday February 28 2008 We will grade Exam 1 this weekend and the grades will be distributed via email on Monday The exam will be returned to you during workshops next week Please carefully look at the exam and if you made any mistakes try to understand why what you did was not correct If you disagree with the grade you received you need to come and talk to me Your TAs can not change your exam grade The next exam will take place on March 25 Please do not wait until the day before the exam to start studying Frank L H Wolfs Department of Physics and Astronomy University of Rochester 2 Physics 121 Quiz lecture 12 The quiz today will have 3 questions and provide me with feedback on Exam 1 Note each answer you submit is correct Frank L H Wolfs Department of Physics and Astronomy University of Rochester The Center of Mass Up to now we have ignored the shape of the objects were are studying Objects that are not point like appear to carry out more complicated motions than pointlike objects e g the object may be rotating during its motion We will find that we can use whatever we have learned about motion of point like objects if we consider the motion of the centerof mass of the extended object Frank L H Wolfs Department of Physics and Astronomy University of Rochester But what is the center of mass and where is it located We will start with considering one dimensional objects For an object consisting out of two point masses the center of mass is defined as xcm m1x1 m2 x2 1 mi xi m1 m2 M i d m2 m1 Xcm Frank L H Wolfs Department of Physics and Astronomy University of Rochester 3 But what is the center of mass and where is it located Let s look at this particular system Since we are free to choose our coordinate system in a way convenient to us we choose it such that the origin coincides with the location of mass m1 The center of mass is located at xcm d m2 m1 Xcm m2 d m1 m2 Note the center of mass does not need to be located at a position where there is mass Frank L H Wolfs Department of Physics and Astronomy University of Rochester But what is the center of mass and where is it located In two or three dimensions the calculation of the center of mass is very similar except that we need to use vectors If we are not dealing with discreet point masses we need to replace the sum with an integral 1 rcm rdm M V Frank L H Wolfs Department of Physics and Astronomy University of Rochester But what is the center of mass and where is it located We can also calculate the position of the center of mass of a two or three dimensional object by calculating its components separately 1 mi xi M i 1 ycm my M i i xcm i zcm 1 mi zi M i Note the center of mass may be located outside the object Frank L H Wolfs Department of Physics and Astronomy University of Rochester 4 Calculating the position of the center of mass Example problem Consider a circular metal plate of radius 2R from which a disk if radius R has been removed Let us call it object X Locate the center of mass of object X Mass M 4 a y axis b y axis x axis x axis Since this object is complicated we can simplify our life by using the principle of superposition If we add a disk of radius R located at R 2 0 we obtain a solid disk of radius R The center of mass of this disk is located at 0 0 Frank L H Wolfs Mass M solid disk Mass 3M 4 Department of Physics and Astronomy University of Rochester Calculating the position of the center of mass Example problem The center of mass of the solid disk can be expressed in terms of the disk X and the disk D we used to fill the hole in disk X xcm C xcm X mX xcm D mD 0 mX mD a This equation can be rewritten as xcm X Mass M 4 y axis b x axis x m RmD cm D D mX mX Where we have used the fact that the center of mass of disk D is located at R 2 0 Thus xcm X R 3 Frank L H Wolfs Mass 3M 4 y axis x axis Mass M solid disk Department of Physics and Astronomy University of Rochester Motion of the center of mass To examine the motion of the center of mass we start with its position and then determine its velocity and acceleration Mrcm mi ri i Mvcm mi vi i Macm mi ai i Frank L H Wolfs Department of Physics and Astronomy University of Rochester 5 Motion of the center of mass The expression for Macm can be rewritten in terms of the forces on the individual components dP d Macm Mvcm cm Fi Fnet ext dt dt i We conclude that the motion of the center of mass is only determined by the external forces Forces exerted by one part of the system on other parts of the system are called internal forces According to Newton s third law the sum of all internal forces cancel out for each interaction there are two forces acting on two parts they are equal in magnitude but pointing in an opposite direction and cancel if we take the vector sum of all internal forces Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion of the center of mass and linear momentum …