Chapter 9 Rotation and Rolling I Rotational variables Angular position displacement velocity acceleration II Rotation with constant angular acceleration III Relation between linear and angular variables Position speed acceleration IV Kinetic energy of rotation V Rotational inertia VI Torque VII Newton s second law for rotation VIII Work and rotational kinetic energy IX Rolling motion I Rotational variables Rigid body body that can rotate with all its parts locked together and without shape changes Rotation axis every point of a body moves in a circle whose center lies on the rotation axis Every point moves through the same angle during a particular time interval Reference line fixed in the body perpendicular to the rotation axis and rotating with the body Angular position the angle of the reference line relative to the positive direction of the x axis arc length radius s r Units radians rad 2 r 2 rad r 1 rad 57 3 0 159 rev 1 rev 360 Note we do not reset to zero with each complete rotation of the reference line about the rotation axis 2 turns 4 Translation body s movement described by x t Rotation body s movement given by t angular position of the body s reference line as function of time Angular displacement body s rotation about its axis changing the angular position from 1 to 2 2 1 Clockwise rotation negative Counterclockwise rotation positive Angular velocity Average Instantaneous Units rad s or rev s avg 2 1 t2 t1 t d t 0 t dt lim These equations hold not only for the rotating rigid body as a whole but also for every particle of that body because they are all locked together Angular speed magnitude of the angular velocity Angular acceleration Average Instantaneous avg 2 1 t 2 t1 t d t 0 t dt lim Angular quantities are normally vector quantities right hand rule Examples angular velocity angular acceleration Object rotates around the direction of the vector a vector defines an axis of rotation not the direction in which something is moving Angular quantities are normally vector quantities right hand rule Exception angular displacements The order in which you add two angular displacements influences the final result is not a vector II Rotation with constant angular acceleration Linear equations v v0 at 1 x x0 v0t at 2 2 v 2 v02 2a x x0 1 x x0 v0 v t 2 1 x x0 vt at 2 2 Angular equations 0 t 1 2 2 02 2 0 0 0 t t 2 1 2 0 0 t 1 2 0 t t 2 III Relation between linear and angular variables Position Speed s r always in radians ds d r v r dt dt in rad s v is tangent to the circle in which a point moves Since all points within a rigid body have the same angular speed points located at greater distance with respect to the rotational axis have greater linear or tangential speed v If constant v constant each point within the body undergoes uniform circular motion Period of revolution T 2 r v 2 r r 2 Acceleration dv d r d r r at r dt dt dt Responsible for changes in the magnitude of the linear velocity vector v Tangential component of linear acceleration 2 v Radial component of ar 2 r r linear acceleration Units m s2 Responsible for changes in the direction of the linear velocity vector v IV Kinetic energy of rotation Reminder Angular velocity is the same for all particles within the rotating body Linear velocity v of a particle within the rigid body depends on the particle s distance to the rotation axis r K 1 2 1 2 1 2 1 1 1 mv1 mv2 mv3 mi vi2 mi ri 2 mi ri2 2 2 2 2 2 i i 2 i 2 Rotational inertia Moment of inertia I Indicates how the mass of the rotating body is distributed about its axis of rotation The moment of inertia is a constant for a particular rigid body and a particular rotation axis Example long metal rod I mi ri2 i Units kg Smaller rotational inertia in a easier to rotate m2 Kinetic energy of a body in pure rotation K 1 2 I 2 Kinetic energy of a body in pure translation K 1 MvCOM 2 2 V Rotational inertia Discrete rigid body I miri2 Continuous rigid body I r2 dm Parallel axis theorem I I COM Mh 2 R h perpendicular distance between the given axis and axis through COM Rotational inertia about a given axis Rotational Inertia about a parallel axis that extends trough body s Center of Mass Mh2 Proof I r 2 dm x a 2 y b 2 dm x 2 y 2 dm 2a xdm 2b ydm a 2 b 2 dm I R 2 dm 2aMxCOM 2bMyCOM Mh 2 I COM Mh 2 VI Torque Torque Twist Turning action of force F Radial component Fr does not cause rotation pulling a door parallel to door s plane Tangential component Ft does cause rotation pulling a door perpendicular to its plane Ft F sin Units Nm r F sin r Ft r sin F r F r Moment arm of F Vector quantity r Moment arm of Ft Sign Torque 0 if body rotates counterclockwise Torque 0 if clockwise rotation Superposition principle When several torques act on a body the net torque is the sum of the individual torques VII Newton s second law for rotation F ma I Proof Particle can move only along the circular path only the tangential component of the force Ft tangent to the circular path can accelerate the particle along the path F t mat Ft r mat r m r r mr 2 I net I VIII Work and Rotational kinetic energy Translation K K f K i 1 2 1 2 mv f mvi W 2 2 Rotation K K f K i xf 1 2 1 2 I f I i W 2 2 f W Fdx W d xi Work kinetic energy Theorem Work rotation about fixed axis i Work constant torque W f i W F d dW P F v dt P dW dt Power rotation about fixed axis Proof W K K f K i 1 2 1 2 1 1 1 1 1 1 mv f mvi m f r 2 m i r 2 mr 2 2f mr 2 i2 I 2f I i2 2 2 2 2 2 2 2 2 f dW Ft ds Ft r d d W d i P dW d dt dt IX Rolling Rotation Translation combined Example bicycle s wheel s R ds d R R vCOM dt dt Smooth rolling motion The motion of any round body rolling smoothly over a surface can be separated into purely rotational and purely translational motions Pure rotation Rotation axis through point where wheel contacts ground Angular speed about P Angular speed about O for stationary observer vtop 2 R 2 R 2vCOM Instantaneous velocity vectors sum of translational and rotational motions Kinetic energy of rolling I p I COM …