Physics 121 March 25 2008 Rotational Motion and Angular Momentum Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 March 25 2008 Course Information Topics to be discussed today Review of Rotational Motion Rolling Motion Angular Momentum Conservation of Angular Momentum Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 March 25 2008 Homework set 7 is now available and is due on Saturday April 5 at 8 30 am There will be no workshops and office hours for the rest of the week We will be busy grading exam 2 The grades for exam 2 will be distributed via email on Monday March 31 You should pick up your exam in workshop next week Please check it carefully for any errors Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Quiz lecture 17 The quiz today will have 4 questions Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational variables A quick review The variables that are used to describe rotational motion are Angular position Angular velocity d dt Angular acceleration d dt The rotational variables are related to the linear variables Linear position l R Linear velocity v R Linear acceleration a R Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational Kinetic Energy A quick review Since the components of a rotating object have a non zero linear velocity we can associate a kinetic energy with the rotational motion 1 1 1 1 2 2 2 2 2 K mi vi mi ri mi ri I 2 i 2 i 2 i 2 The kinetic energy is proportional to square of the rotational velocity Note the equation is similar to the translational kinetic energy 1 2 mv2 except that instead of being proportional to the the mass m of the object the rotational kinetic energy is proportional to the moment of inertia I of the object Note units of I kg m2 I mi ri 2 or I r 2 dm i Frank L H Wolfs Department of Physics and Astronomy University of Rochester Torque In general the torque associated with a force F is equal to rF sin r F The arm of the force also called the moment arm is defined as rsin The arm of the force is the perpendicular distance of the axis of rotation from the line of action of the force If the arm of the force is 0 the torque is 0 and there will be no rotation The maximum torque is achieved when the angle is 90 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Torque A quick review The torque of the force F is related to the angular acceleration F This equation looks similar to Newton s second law for linear motion r A F ma Note linear mass m force F Frank L H Wolfs rotational moment I torque Department of Physics and Astronomy University of Rochester Rolling motion A quick review Rolling motion is a combination of translational and rotational motion The kinetic energy of rolling motion has thus two contributions Translational kinetic energy 1 2 M vcm2 Rotational kinetic 1 2 Icm 2 energy Assuming the wheel does not slip v R Frank L H Wolfs Department of Physics and Astronomy University of Rochester How different is a world with rotational motion Sample problem Consider the loop to loop What height h is required to make it to the top of the loop First consider the case without rotation Initial mechanical energy mgh Minimum velocity at the top of the loop is determine by requiring that mv2 R mg or v2 gR The mechanical energy is thus equal to 1 2 mv2 2mgR 5 2 mgR Conservation of energy requires h 5 2 R Frank L H Wolfs h R Department of Physics and Astronomy University of Rochester How different is a world with rotational motion Sample problem What changes when the object rotates The minimum velocity at the top of the loop will not change The minimum translational kinetic energy at the top of the loop will not change But in addition to translational kinetic energy there is now also rotational kinetic energy The minimum mechanical energy is at the top of the loop has thus increased The required minimum height must thus have increased h R OK let s now calculate by how much the minimum height has increased Frank L H Wolfs Department of Physics and Astronomy University of Rochester How different is a world with rotational motion Sample problem The total kinetic energy at the top of the loop is equal to Kf 1 2 1 1 I I Mv 2 2 M v 2 2 2 2 r This expression can be rewritten as 1 2 2 7 Kf M M v Mv 2 2 5 10 We now know the minimum mechanical energy required to reach this point and thus the minimum height h Frank L H Wolfs 27 R 10 h R Note without rotation h 25 10 R Department of Physics and Astronomy University of Rochester Torque and rotational motion Let s test our understanding of the basic aspects of torque and rotational motion by working on the following concept problems Q17 1 Q17 2 Q17 3 Q17 4 Q17 5 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Torque The torque associated with a force is a vector It has a magnitude and a direction The direction of the torque can be found by using the right hand rule to evaluate r x F For extended objects the total torque is equal to the vector sum of the torque associated with each component of this object Frank L H Wolfs Department of Physics and Astronomy University of Rochester Angular momentum We have seen many similarities between the way in which we describe linear and rotational motion Our treatment of these types of motion are similar if we recognize the following equivalence linear rotational mass m moment I force F torque r x F What is the equivalent to linear momentum Answer angular momentum Frank L H Wolfs Department of Physics and Astronomy University of Rochester Angular momentum The angular momentum is defined as the vector product between the position vector and the linear momentum Note Compare this definition with the definition of the torque Angular momentum is a vector The unit of angular momentum is kg m2 s The angular momentum depends on both the magnitude and the direction of the position and linear momentum vectors Under certain circumstances the angular momentum of a system is conserved Frank L H Wolfs Department of Physics and Astronomy University of Rochester Angular momentum Consider an object carrying out circular motion For this type of motion the position vector will be perpendicular to the momentum vector The magnitude of the angular momentum is equal to the product of the magnitude of the radius r and the linear momentum p L mvr mr2 v r I Note