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Chapter 10: Rotational Motion

Chapter 10: 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.

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Chapter 10: Rotational Motion

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  1. Chapter 10: 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 - 9:Methods to analyze the dynamics of objects in TRANSLATIONAL MOTION. Newton’s Laws! • Chs. 4 - 6: Newton’s Laws using Forces • Chs. 7 & 8: Newton’s Laws using Energy & Work • Ch. 9: Newton’s Laws using Momentum. NOW • Chs. 10 & 11: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: 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) 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.

  8. Sect. 10.1: 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!

  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  Reference Line

  10. 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 ℓ 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 lengthℓisdefinedas: θ (ℓ/r) • Conversion between radians & degrees: θfor a full circle = 360º = (ℓ/R) radians Arc lengthℓfor 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

  12. Angular Displacement: Ex. 10-1: Birds of prey—in radians A particular bird’s eye can just distinguish objects that subtend an angle no smaller than about θ = 3  10-4 rad. a. How many degrees is this? b.How small an object can the bird just distinguish when flying at a height of 100 m?

  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!

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