1. Chapter 8: Rotational Motion

2. Topic of Chapter: Objects rotating –First, rotating, without translating. –Then, rotating AND translating together. Assumption: Rigid Body –Definite shape. Does not deform or change shape. Rigid Body motion = Translational motion of center of mass (everything done up to now) + Rotational motion about an axis through center of mass. Can treat the two parts of motion separately.

3. COURSE THEME: NEWTON’S LAWS OF MOTION! Chs. 4 - 7: Methods to analyze the dynamics of objects in TRANSLATIONAL MOTION. Newton’s Laws! –Chs. 4 & 5: Newton’s Laws using Forces –Ch. 6: Newton’s Laws using Energy & Work –Ch. 7: Newton’s Laws using Momentum. NOW Ch. 8: Methods to analyze dynamics of objects in ROTATIONAL LANGUAGE. Newton’s Laws in Rotational Language! –First, Rotational Language. Analogues of each translational concept we already know! –Then, Newton’s Laws in Rotational Language.

4. A rigid body is an extended object whose size, shape, & distribution of mass don’t change as the object moves and rotates. Example: a CD  Rigid Body Rotation

5. Three Basic Types of Rigid Body Motion

6. Pure Rotational Motion All points in the object move in circles about the rotation axis (through the Center of Mass) Reference Line The axis of rotation is through O & is  to the picture. All points move in circles about O r

7. In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time. r r

8. Sect. 8-1: Angular Quantities Description of rotational motion: Need concepts: Angular Displacement Angular Velocity, Angular Acceleration Defined in direct analogy to linear quantities. Obey similar relationships! Positive Rotation! r

9. Rigid object rotation: –Each point (P) moves in a circle with the same center! Look at OP: When P (at radius R) travels an arc length ℓ, OP sweeps out angle θ. θ  Angular Displacement of the object  Reference Line r

10. θ  Angular Displacement Commonly, measure θ in degrees. Math of rotation: Easier if θ is measured in Radians 1 Radian  Angle swept out when the arc length = radius When  R, θ  1 Radian θ in Radians is defined as: θ = ratio of 2 lengths (dimensionless) θ MUST be in radians for this to be valid!  Reference Line r

11. θ in Radians for a circle of radius r, arc length is defined as: θ  ( /r) Conversion between radians & degrees: θ for a full circle = 360º = ( /r) radians Arc length for a full circle = 2πr  θ for a full circle = 360º = 2π radians Or 1 radian (rad) = (360/2π)º  57.3º Or 1º = (2π/360) rad  0.017 rad –In doing problems in this chapter, put your calculators in RADIAN MODE!!!!

12. Example 8-2: θ  3  10 -4 rad = ? º r = 100 m, = ? a) θ = (3  10 -4 rad)  [(360/2π)º/rad] = 0.017º b) = rθ = (100)  (3  10 -4 ) = 0.03 m = 3 cm θ MUST be in radians in part b!

13. Angular Displacement

14. Average Angular Velocity = angular displacement  θ = θ 2 – θ 1 (rad) divided by time  t: (Lower case Greek omega, NOT w!) Instantaneous Angular Velocity (Units = rad/s) The SAME for all points in the object! Valid ONLY if θ is in rad! Angular Velocity (Analogous to linear velocity!)

15. Average Angular Acceleration = change in angular velocity  ω = ω 2 – ω 1 divided by time  t: (Lower case Greek alpha!) Instantaneous Angular Acceleration = limit of α as  t,  ω  0 (Units = rad/s 2 ) The SAME for all points in body! Valid ONLY for θ in rad & ω in rad/s! Angular Acceleration (Analogous to linear acceleration!)

16. Ch. 5 (circular motion): A mass moving in a circle has a linear velocity v & a linear acceleration a. We’ve just seen that it also has an angular velocity & an angular acceleration.  There MUST be relationships between the linear & the angular quantities! Relations of Angular & Linear Quantities Δθ Δ r

17. Connection Between Angular & Linear Quantities v = (  /  t),  = r  θ  v = r(  θ/  t) = rω Radians!  v = rω  Depends on r (ω is the same for all points!) v B = r B ω B, v A = r A ω A v B > v A since r B > r A

18. Summary: Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related as:

19. Relation Between Angular & Linear Acceleration In direction of motion: (Tangential acceleration!) a tan = (  v/  t),  v = r  ω  a tan = r (  ω/  t) a tan = rα a tan : depends on r α : the same for all points _____________

20. Angular & Linear Acceleration From Ch. 5: there is also an acceleration  to the motion direction (radial or centripetal acceleration) a R = (v 2 /r) But v = rω  a R = r ω 2 a R : depends on r ω: the same for all points _____________

21. Total Acceleration  Two  vector components of acceleration Tangential: a tan = rα Radial: a R = rω 2 Total acceleration = vector sum: a = a R + a tan _____________ a  ---

22. Relation Between Angular Velocity & Rotation Frequency Rotation frequency: f = # revolutions / second (rev/s) 1 rev = 2π rad  f = (ω/2π) or ω = 2π f = angular frequency 1 rev/s  1 Hz (Hertz) Period: Time for one revolution.  T = (1/f) = (2π/ω)

23. Translational-Rotational Analogues & Connections ANALOGUES Translation Rotation Displacementx θ Velocityv ω Accelerationa α CONNECTIONS = rθ, v = rω a tan = r α a R = (v 2 /r) = ω 2 r

24. Correspondence between Linear & Rotational quantities

25. On a rotating merry-go-round, one child sits on a horse near the outer edge & another child sits on a lion halfway out from the center. a. Which child has the greater translational velocity v? b. Which child has the greater angular velocity ω? Conceptual Example 8-3: Is the lion faster than the horse?

26. Example 8-4: Angular & Linear Velocities & Accelerations A merry-go-round is initially at rest (ω 0 = 0). At t = 0 it is given a constant angular acceleration α = 0.06 rad/s 2. At t = 8 s, calculate the following: a. The angular velocity ω. b. The linear velocity v of a child located r = 2.5 m from the center. c. The tangential (linear) acceleration a tan of that child. d. The centripetal acceleration a R of the child. e. The total linear acceleration a of the child.

27. Example 8-5: Hard Drive The platter of the hard drive of a computer rotates at frequency f = 7200 rpm (rpm = revolutions per minute = rev/min) a. Calculate the angular velocity ω (rad/s) of the platter. b. The reading head of the drive r = 3 cm (= 0.03 m) from the rotation axis. Calculate the linear speed v of the point on the platter just below it. c. If a single bit requires 0.5 μm of length along the direction of motion, how many bits per second can the writing head write when it is r = 3 cm from the axis?