1 / 27

Chapter 8: Rotational Motion

Chapter 8: Rotational Motion. 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.

vivienne-dotson
Download Presentation

Chapter 8: Rotational Motion

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 inROTATIONAL 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. Rigid Body Rotation 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

  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) r Reference Line The axis of rotation is through O & is  to the picture. All points move in circles about O

  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! r Positive Rotation!

  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 Displacementof the object r  Reference Line

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

  11. θin Radians for a circle of radius r, arc length  isdefinedas: θ (/r) • Conversion between radians & degrees: θfor a full circle = 360º = (/r) radians Arc length for a full circle = 2πr  θfor a full circle = 360º = 2πradians 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 θMUST be in radians in part b!

  13. Angular Displacement

  14. Angular Velocity(Analogous to linear velocity!) Average Angular Velocity = angular displacement θ = θ2 – θ1 (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 θis in rad!

  15. Angular Acceleration(Analogous to linear acceleration!) • Average Angular Acceleration= change in angular velocity ω = ω2– ω1 divided by time t: (Lower case Greek alpha!) • Instantaneous Angular Acceleration = limit of α as t, ω0 (Units = rad/s2) TheSAMEfor all points in body! Valid ONLYfor θin rad & ω in rad/s!

  16. Relations of Angular & Linear Quantities 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. Δ Δθ r  There MUST be relationships between the linear & the angular quantities!

  17. Connection Between Angular & Linear Quantities Radians!  v = (/t),  = rθ  v = r(θ/t) = rω v = rω Depends on r (ω is the same for all points!) vB = rBωB,vA = rAωA vB > vA since rB > rA

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

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

  20. Angular & LinearAcceleration _____________ From Ch. 5: there is also an acceleration  to the motion direction (radial or centripetal acceleration) aR = (v2/r) But v = rω  aR= rω2 aR: depends on r ω: the same for all points

  21. Total Acceleration _____________  Two vector components of acceleration • Tangential: atan= rα • Radial: aR= rω2 • Total acceleration = vector sum: a = aR+ atan a ---

  22. Relation Between Angular Velocity & Rotation Frequency • Rotation frequency: f = # revolutions / second (rev/s) 1 rev = 2πrad  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 Displacement x θ Velocity v ω Acceleration a α CONNECTIONS  = rθ, v = rω atan= r α aR = (v2/r) = ω2 r

  24. Correspondence between Linear & Rotational quantities

  25. Conceptual Example 8-3: Is the lion faster than the horse? 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?

  26. Example 8-4: Angular & Linear Velocities & Accelerations A merry-go-round is initially at rest (ω0 = 0). At t = 0 it is given a constant angular acceleration α = 0.06 rad/s2. 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 atan of that child. d. The centripetal acceleration aR 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 velocityω(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?

More Related