Physics 121 March 18 2008 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 March 18 2008 Course Information Topics to be discussed today Variables used to describe rotational motion The equations of motion for rotational motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Course Announcements Homework set 6 is now available on the WEB and will be due on Saturday morning March 22 at 8 30 am All the material to be covered on Exam 2 has now been discussed Today we will start on material that will be covered on Exam 3 Exam 2 will take place on Tuesday March 25 at 8 am in Hubbell It will cover the material discussed in Chapters 7 8 and 9 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Homework Set 6 Calculating the moment of inertia Equations of motion with constant Equations of motion with constant Rotational kinetic energy Note this is not a rigid body Frank L H Wolfs Torque and acceleration Department of Physics and Astronomy University of Rochester Rotational motion variables In our discussion of rotational motion we will first focus on the rotation of rigid objects around a fixed axis The variables that are used to describe this type of motion are similar to those we use to describe linear motion Angular position Angular velocity Angular acceleration Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion variables The angular position measured in radians is the angle of rotation of the object with respect to a reference position The angular position of point P at this point in time is equal to In order to uniquely define this position we have assume that an the angular position is measured with respect to the x axis Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion variables Note The angular position is always specified in radians One radian is the angular displacement corresponding to a linear displacement l R Make sure you keep track of the sign of the angular position An increase in the angular position corresponds to a counterclockwise rotation a decrease corresponds to a clockwise rotation Frank L H Wolfs Department of Physics and Astronomy University of Rochester Complex motion in Cartesian coordinates is simple motion in rotational coordinates Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion variables If we look at an object carrying out a rotation around a fixed axis we will see that the angular position becomes a function of time To describe the rotational motion we introduce the concepts of angular velocity and angular acceleration Remember for linear motion we found it useful to introduce the concepts of linear velocity and linear acceleration Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion variables For both velocity and acceleration we can talk about instantaneous and average Angular velocity Definition d dt Symbol Units rad s Angular acceleration Definition d dt d2 dt2 Symbol Units rad s2 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion constant acceleration If the object experiences a constant angular acceleration then we can describe its rotational motion with the following equations of motion t 0 t 1 2 t 0 0t t 2 Note how similar these equations are to the equation of motion for linear motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion constant acceleration Example problem A wheel starting from rest rotates with a constant angular acceleration of 2 0 rad s2 During a certain 3 0 s interval it turns through 90 rad a How long had the wheel been turning before the start of the 3 0 s interval b What was the angular velocity of the wheel at the start of the 3 0 s interval Define time t 0 s as the time that the wheel is at rest The angular velocity and the angle of rotation at a later time t are given by 1 2 t t t t 2 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion constant acceleration Example problem The change in the angular position during a time period t can now be calculated 1 1 2 1 2 2 t t t t t t t t t t 2 2 2 Since the problem specifies and t we can now calculate the time t and the angular velocity at that time 1 1 2 2 t t 1 2 2 t t t t t t 2 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion variables The linear velocity of a part of the rigid body is related to the angular velocity of the object Consider point P It this point makes one complete revolution it travels a distance 2 R When the angular position changes by d point P moves a distance dl 2 R d 2 Rd The linear velocity of point P is equal to v dl dt Rd dt R Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rotational motion variables Things to consider when looking at the rotation of rigid objects around a fixed axis Each part of the rigid object has the same angular velocity Only those parts that are located at the same distance from the rotation axis have the same linear velocity The linear velocity of parts of the rigid object increases with increasing distance from the rotation axis Frank L H Wolfs Department of Physics and Astronomy University of Rochester Relation between rotational and linear variables Although in rotational motion we prefer to use rotational variables we can also express the motion in terms of linear variables s r ds v dt dv at dt Frank L H Wolfs d d r r r dt dt d d r r r dt dt Department of Physics and Astronomy University of Rochester Rotational motion acceleration Note the acceleration at r is only one of the two component of the acceleration of point P The two components of the acceleration of point P are The radial component this component is always present since point P carried out circular motion around the axis of rotation The tangential component this component is present only when the angular acceleration is not equal to 0 rad s2 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Rolling motion To describe rolling motion we need to use both translational and rotational motion The rolling motion can be described in terms of pure rotational motion with respect to the contact point P which is always at rest The rotation axis around P is called the