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Chapter 8: Rotational Motion

Chapter 8: Rotational Motion. Topics of Chapter: Objects Rotating Newton’s Laws of Motion applied to rotating objects , translated into rotational language . First, rotating, without translating. Then, rotating AND translating together. Assumption: Rigid Bodies

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Chapter 8: Rotational Motion

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  1. Chapter 8: Rotational Motion

  2. Topics of Chapter:Objects Rotating • Newton’s Laws of Motion applied to rotating objects, translated into rotational language. • First, rotating, without translating. • Then, rotating AND translating together. • Assumption:Rigid Bodies • 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! Everything up to now: • Methods to analyze the dynamics of objects in TRANSLATIONAL MOTION. Newton’s Laws! • Chs. 3, 4, 5: Newton’s Laws using Forces • Ch. 6: Newton’s Lawsusing Energy & Work • Ch. 7: Newton’s Laws using Momentum. NOW • Ch. 8: Methods to analyze dynamics of objects in ROTATIONAL LANGUAGE.

  4. Course Theme: Newton’s Laws of Motion! • Ch. 8: Methods to analyze dynamics of objectsin ROTATIONAL LANGUAGE. Newton’s Laws in Rotational Language! • First, introduction to Rotational Language. • Then, Rotational Analoguesof each translational concept we already know! • Then, Newton’s Laws in Rotational Language.

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

  6. There are 3 Basic Types of Rigid Body Motion

  7. 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 aboutO

  8. Pure Rotational Motion • All points on objectmove • in circles around the • axis of rotation (“O”). • The circle radius is r. • All points on a straight • line drawn through the • axis move through the • same angle in the • same time. r r

  9. Angular Quantities r • To describe rotational motion, we need Rotational Concepts: Angular Displacement,Angular Velocity, Angular Acceleration • These are defined in direct analogyto linear quantities & they obey similar relationships! Positive Rotation!

  10. Rigid Body Rotation • Each point (P) • moves in a circle • with the same center! • Look at OP: • When P (at radius • r) sweeps out angle θ. • θ Angular Displacement • of the object r

  11. θ Angular Displacement • Commonly, we 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! r  Reference Line   (/r)

  12. 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!!!!

  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 Between 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: 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: 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: 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?

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