1. 1 Rotational Kinematics Chapter 9 October 11, 2005 Today’s Topics Translation vs. rotation Variables used for rotation: , ,  Four angular equations of motion for  = constant Relation between linear and angular variables Introduction to rolling Reminder: “Kinematics” is the description of motion Linear motion: we have used position (x), velocity and speed (v), acceleration (a), and time (t) Rotational motion: we will use time and three new variables: , , 

2. 2 Translation vs. Rotation Pure translation: every part of an object undergoes the same displacement as the object moves; it maintains the same orientation Pure rotation: every part of an object moves in a circle with centers lying along a line – the axis of rotation Examples or pure rotation: Twirl a mass in a horizontal circle above your head Stationary, spinning bicycle wheel Door rotating on hinges Combination of translation and rotation Examples: Any rolling object (wheel, ball, ….) Boomerang, Frisbee Tennis ball served with spin, curve ball Diver doing a forward “three and a half” We first focus on pure rotation only, then we combine rotation with translation.

3. 3 x y r  Variables Used in Rotation Imagine a mass attached to a string, moving in a circle as shown One can locate the mass in the x,y plane Using Cartesian coordinates (x,y) Using polar coordinates (r,  ) where  is measured from the +x axis Positive  if measured counterclockwise from +x axis Negative  if measured clockwise from +x axis

4. 4 Variables Used in Rotation Our standard unit for  is radians, but  is also measured in degrees and revolutions Radians Degrees Revolutions 2  360 o 1 rev x y r  s = arc length along circle Definition of radians  (in radians) = s/r = arc length/radius  = 360 o   (rad) = 2  r/r = 2  rad 1 radian = 57.3 o

5. 5 Variables Used in Rotation x y r   Angular velocity, , is how  changes with time:  is the Greek letter omega Units for  : rad/sec, rev/sec You are familiar with 3000 “rpm” (revolutions per minute): car tachometer +  is counterclockwise rotation -  is clockwise rotation Angular acceleration, , is how  changes with time:  is Greek letter alpha Units for : rad/sec 2, rev/sec 2

6. 6 Variables Used in Rotation Note about angular variables Refer to stationary bicycle wheel spinning Linear velocity of parts depends on radius Rim moves fast, hub moves slowly Angular quantities are properties of the whole rotating body Example: all bicycle wheel spokes sweep out the same slice of  in a given time

7. 7 4 Angular equations of motion for  = constant x x y y vxvx r   Translation Rotation x  v  a  For a = constant For  = constant See handout on “Rotational Kinematics”

8. 8 Problem Example A spin drier initially is turning at 20.0 rad/s and slows down uniformly to 10.0 rad/s in 50 revolutions. The drier is a cylinder with radius of 0.30 m. a)What is the angular acceleration? b)What is the time required for the 50 revolutions? c)What are the radial and tangential acceleration on the side of the drier as it begins to slow down?

9. 9 Relations between translational and rotational variables From definition of angle  in radians,  (in radians) = s/r, we have: Arc length s =  r Now assume object is rotating counter- clockwise with a constant  -- next page x y r  s = arc length along circle

10. 10 Relations between translational and rotational variables From arc length s =  r, ds/dt = Tangential velocity v T = (d  /dt) r =  r So the tangential velocity depends on radius for a given . Now assume that  is increasing, so that we have a nonzero  -- next page. x y r  s = arc length  Tangential velocity

11. 11 Relations between translational and rotational variables We already know that any rotating object is accelerating INWARD (centripetal acceleration), but if there is angular acceleration, the object is also accelerating FORWARD (tangential acceleration) From tangential velocity v T =  r, we have tangential acceleration a T = dv T /dt = (d  /dt) r =  r As promised, let’s now derive the expression for centripetal acceleration a C = v T 2 /r =  2 r which we have so far just been assuming to be true. x y   Tangential velocity Tangential acceleration, a T Centripetal acceleration, a C

12. 12 Derivation of a c = v 2 /r x y r  Let  = constant so that  =  t, i.e. particle rotates around circle with constant speed. At a given , the x,y coordinates of the particle are: Cosine and sine go between ±1, so the x and y coordinates go between ±r as expected. Differentiate x(t) and y(t) to find v x (t) and v y (t) --- next page x y

13. 13 Differentiating v x (t) and v y (t) yields a x (t) and a y (t): Derivation of a c = v 2 /r x y Differentiating x(t) and y(t) yields: axax ayay The magnitude of the centripetal acceleration vector is then: Direction of acceleration is radially inward as shown in diagram.

14. 14 Example of a rotating object which is accelerating with both centripetal and tangential acceleration Simple pendulum swinging back and forth  Tangential acceleration Centripetal acceleration Tangential velocity (increasing) Net acceleration vector Pivot point

15. 15 Rolling r v cm v cm =  r  Goal: relate a rolling object’s linear speed (v cm ) to how fast it turns (  ) Consider wheel rolling (without slipping or skidding) through one revolution in time T Covers distance d = 2  r in time T