PH 201 Dr. Cecilia Vogel Lecture 17

REVIEW  CM with holes  Thrust OUTLINE  Rotational Motion  angular displacement  angular velocity  angular acceleration

Rigid Body  Rigid means the object keeps its size and shape, all parts staying in same relative position.  water-balloons are NOT rigid  they don’t keep size/shape  cars are not rigid  wheels rotate, rest of car doesn’t

Rotation of a Rigid Body  As a rigid body rotates,  all parts rotate through same angle  notice that the parts close to the center don’t move as far  but the angles are the same

Angular Displacement  The angular displacement is the change in angular position  Counterclockwise usually is taken as +  clockwise as -  choose an x -axis to measure your angles from, if necessary

Distance Traveled  The distance a point travels  equals the arclength  which is proportional to distance from center of rotation s r

Units?  What are the units of angular displacement?  if s=r   then the units of  should be  units of s/units of r = m/m = unitless  The angular displacement is unitless  However,  since we sometimes measure in degrees or revolutions  we need to keep track (placeholder)  we say we measure angles in radians

Radians  Once around the circle  s = 2  r  so  2  rad is one complete revolution.  Conversion factors:  2  rad = 360 degrees = 1 rev

Example Bicycle wheel rotates ¼ turn What is the angular displacement in rad? How far has a point at the center moved? How far has a point 20 cm from the center moved?

Start a Table Linear variable Angular variable Variable name x v=dx/dt a=dv/dt F=ma m K= ½ mv 2 p=mv

Angular Velocity  Angular velocity,  is rate of change of angle  =d  /dt  Average angular velocity is  (Instantaneous) angular velocity is

Angular Acceleration  Angular acceleration,  is the rate of change of angular velocity  Average angular acceleration is  (Instantaneous) angular acceleration is

Sign of Angular Variables  The angular displacement and angular velocity are taken as positive  counterclockwise.  The angular acceleration is positive  if angular velocity increases  starting or speeding up counterclockwise  OR slowing down or stopping clockwise  The angular acceleration is negative  if angular velocity decreases  starting or speeding up clockwise  OR slowing down or stopping counterclockwise

Angular Graph  On a graph of angle vs. time,  slope is angular velocity.  curvature is angular acceleration.  On a graph of angular velocity vs. time,  slope is angular acceleration  t

Constant Angular Velocity  If the angular velocity,  is constant,     vs. t is a linear function

Constant Angular Acceleration  If the angular acceleration,  is constant,   and the angle is a  also

Newton  Forms of two of Newton’s Laws apply to angular motion:  A rigid body that’s not rotating  won’t start rotating,  and a rotating rigid body rotating  continues uniform rotation  unless acted on by an external torque

Torque  An net external torque can change an objects angular velocity  The torque required is proportional to the angular acceleration  The moment of inertia, I, is a measure of how hard it is to change the object’s rotation

Add to Table Linear variable Angular variable Variable name x  angle (rad) v = dx/dt  d  /dt angular velocity (rad/s) a = dv/dt  d  /dt ang. acceleration (rad/s 2 ) F  torque (Nm) mI moment of inertia (kgm 2 ) K p