Copyright © 2012 Pearson Education Inc. PowerPoint ® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Chapter 9 Rotation of Rigid Bodies

Copyright © 2012 Pearson Education Inc. Goals for Chapter 9 To describe rotation in terms of: –angular coordinates (  ) –angular velocity (  ) –angular acceleration (  ) To analyze rotation with constant angular acceleration To relate rotation to the linear velocity and linear acceleration of a point on a body

Copyright © 2012 Pearson Education Inc. Goals for Chapter 9 To understand moment of inertia (I): –how it depends upon rotation axes –how it relates to rotational kinetic energy –how it is calculated

Copyright © 2012 Pearson Education Inc. Introduction A wind turbine, a CD, a ceiling fan, and a Ferris wheel all involve rotating rigid objects. Real-world rotations can be very complicated because of stretching and twisting of the rotating body. But for now we’ll assume that the rotating body is perfectly rigid.

Copyright © 2012 Pearson Education Inc. Angular coordinate Consider a meter with a needle rotating about a fixed axis. Angle  (in radians) that needle makes with +x-axis is a coordinate for rotation.

Copyright © 2012 Pearson Education Inc. Angular coordinates Example: A car’s speedometer needle rotates about a fixed axis. Angle  the needle makes with negative x-axis is the coordinate for rotation. KEY: Define your directions! 

Copyright © 2012 Pearson Education Inc. Units of angles Angles in radians  = s/r. One complete revolution is 360° = 2π radians.

Copyright © 2012 Pearson Education Inc. Angular velocity Angular displacement  of a body is  =  2 –  1 (in radians) Average angular velocity of a body is  av =  /  t (in radians/second)

Copyright © 2012 Pearson Education Inc. Angular velocity The rotation axis matters! Subscript z means that the rotation is about the z-axis. Instantaneous angular velocity is  z = d  /dt. Counterclockwise rotation is positive; Clockwise rotation is negative.

Copyright © 2012 Pearson Education Inc. Calculating angular velocity Flywheel diameter 0.36 m; Suppose  t  = (2.0 rad/s 3 ) t 3

Copyright © 2012 Pearson Education Inc. Calculating angular velocity Flywheel diameter 0.36 m;  = (2.0 rad/s 3 ) t 3 Find  at t 1 = 2.0 s and t 2 = 5.0 s Find distance rim moves in that interval Find average angular velocity in rad/sec & rev/min Find instantaneous angular velocities at t 1 & t 2

Copyright © 2012 Pearson Education Inc. Angular velocity is a vector Angular velocity is defined as a vector whose direction is given by the right-hand rule.

Copyright © 2012 Pearson Education Inc. Angular acceleration Average angular acceleration is  avg-z =  z /  t. Instantaneous angular acceleration is  z = d  z /dt = d 2  /dt 2.

Copyright © 2012 Pearson Education Inc. Angular acceleration Average angular acceleration is  avg-z =  z /  t. Instantaneous angular acceleration is  z = d  z /dt = d 2  /dt 2. For same flywheel with dia = 0.36 m;  t  = (2.0 rad/s 3 ) t 3 find average angular accelerations between t 1 & t 2, & instantaneous accelerations at those t’s

Copyright © 2012 Pearson Education Inc. Angular acceleration as a vector For a fixed rotation axis, angular acceleration  and angular velocity  vectors both lie along that axis.

Copyright © 2012 Pearson Education Inc. Angular acceleration as a vector For a fixed rotation axis, angular acceleration  and angular velocity  vectors both lie along that axis. BUT THEY DON’T HAVE TO BE IN THE SAME DIRECTION!  Speeds up   Slows down 

Copyright © 2012 Pearson Education Inc. Rotation with constant angular acceleration Linear and Angular Motion with constant acceleration equations are very similar!

Copyright © 2012 Pearson Education Inc. Rotation of a Blu-ray disc A Blu-ray disc is coming to rest after being t = 0,  = 27.5 rad/sec;  = rad/s 2 What is  at t = 0.3 seconds? What angle does PQ make with x axis then?

Copyright © 2012 Pearson Education Inc. Relating linear and angular kinematics For a point a distance r from the axis of rotation: its linear speed is v = r  (meters/sec)

Copyright © 2012 Pearson Education Inc. Relating linear and angular kinematics For a point a distance r from the axis of rotation: its linear tangential acceleration is a tan = r  (m/s 2 ) its centripetal (radial) acceleration is a rad = v 2 /r = r 

Copyright © 2012 Pearson Education Inc. An athlete throwing a discus Whirl discus in circle of r = 80 cm; at some time t athlete is rotating at 10.0 rad/sec; speed increasing at 50.0 rad/sec/sec. Find tangential and centripetal accelerations and overall magnitude of acceleration

Copyright © 2012 Pearson Education Inc. Designing a propeller Say rotation of propeller is at a constant 2400 rpm, as plane flies forward at 75.0 m/s at constant speed. But…tips of propellers must move slower than 270 m/s to avoid excessive noise. What is maximum propeller radius? What is acceleration of the tip?

Copyright © 2012 Pearson Education Inc. Designing a propeller Tips of propellers must move slower than than 270 m/s to avoid excessive noise. What is maximum propeller radius? What is acceleration of the tip?

Copyright © 2012 Pearson Education Inc. Moments of inertia How much force it takes to get something rotating, and how much energy it has when rotating, depends on WHERE the mass is in relation to the rotation axis.

Copyright © 2012 Pearson Education Inc. Moments of inertia Getting MORE mass, FARTHER from the axis, to rotate will take more force! Some rotating at the same rate with more mass farther away will have more KE!

Copyright © 2012 Pearson Education Inc. Moments of inertia

Copyright © 2012 Pearson Education Inc. Moments of inertia

Copyright © 2012 Pearson Education Inc. Moments of inertia of some common bodies

Copyright © 2012 Pearson Education Inc. Rotational kinetic energy The moment of inertia of a set of discrete particles is I = m 1 r m 2 r … =  m i r i 2 Rotational kinetic energy of rigid body with moment of inertia I KE(rotation) = 1/2 I  2 (still in Joules!) Since I varies by location, KE varies depending upon axis!

Copyright © 2012 Pearson Education Inc. Rotational kinetic energy example 9.6 What is I about A? What is I about B/C? What is KE if it rotates through A with  =4.0 rad/sec?

Copyright © 2012 Pearson Education Inc. An unwinding cable Wrap a light, non-stretching cable around solid cylinder of mass 50 kg; diameter.120 m. Pull for 2.0 m with constant force of 9.0 N. What is final angular speed and final linear speed of cable?

Copyright © 2012 Pearson Education Inc. More on an unwinding cable Consider falling mass “m” tied to rotating wheel of mass M and radius R What is the resulting speed of the small mass when it reaches the bottom?

Copyright © 2012 Pearson Education Inc. More on an unwinding cable Consider falling mass “m” tied to rotating wheel of mass M and radius R What is the resulting speed of the small mass when it reaches the bottom? Method 1: Energy!

Copyright © 2012 Pearson Education Inc. More on an unwinding cable mgh ½ mv 2 + ½ I  2

Copyright © 2012 Pearson Education Inc. More on an unwinding cable mgh = ½ mv 2 + ½ I  2 I = ½ MR 2 for disk mgh = ½ (m+ ½ M) v 2 V = [2gh/(1+M/2m)] ½

Copyright © 2012 Pearson Education Inc. The parallel-axis theorem What happens if you rotate about an EXTERNAL axis, not internal? Spinning planet orbiting around the Sun Rotating ball bearing orbiting in the bearing Mass on turntable

Copyright © 2012 Pearson Education Inc. The parallel-axis theorem What happens if you rotate about an EXTERNAL axis, not internal? Effect is a COMBINATION of TWO rotations Object itself spinning; Point mass orbiting Net rotational inertia combines both: The parallel-axis theorem is: I P = I cm + Md 2

Copyright © 2012 Pearson Education Inc. The parallel-axis theorem The parallel-axis theorem is: I P = I cm + Md 2. M d

Copyright © 2012 Pearson Education Inc. The parallel-axis theorem The parallel-axis theorem is: I P = I cm + Md 2. Mass 3.6 kg, I = kg-m 2 through P. What is I about parallel axis through center of mass?