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Chapter 10: Rigid Object Rotation

Chapter 10: Rigid Object Rotation. Topic of Chapter: Bodies 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: Rigid Object Rotation

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  1. Chapter 10: Rigid Object Rotation

  2. Topic of Chapter: Bodies 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 axis through center of mass. Can treat two parts of motion separately.

  3. COURSE THEME: NEWTON’S LAWS OF MOTION! • Chs. 5 - 9:Methods to analyze dynamics of objects in TRANSLATIONAL MOTION. Newton’s Laws! • Chs. 5 & 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 do not 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 body move in circles about the rotation axis (through the CM) Reference Line  Axis of rotation through O &  to picture. All points move in circles about O

  7. 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! 

  8. Rigid body rotation: • Each point (P) moves moves in circle with the same center! • Look at OP: When P (at radius r) travels an arc length , OP sweeps out an angle θ. θAngular Displacementof the object  Reference Line NOTE: Your text calls the arc length s instead of !

  9.  here is text’s s! • 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 (/r) θ= ratio of 2 lengths (dimensionless) θMUST be in radians for this to be valid!  Reference Line

  10. 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

  11. Angular Velocity(Analogous to linear velocity!) • Ave. angular velocity = angular displacement θ = θf - θi(rad) divided by timet: ωavg (θ/t) (Lower case Greek omega, NOT w!) • Instantaneous angular velocity = limit ω as t,θ0 ω limt0 (θ/t) = (dθ/dt) (Units = rad/s) The SAMEfor all points in the body! Valid ONLYif θis in rad!

  12. Angular Acceleration(Analogous to linear acceleration!) • Average angular acceleration = change in angular velocity ω = ωf- ωi divided by time t: αavg (ω/t) (Lower case Greek alpha!) • Instantaneous angular accel. = limit of α as t, ω0 α limt0 (ω/t) = (dω/dt) (Units = rad/s2) TheSAMEfor all points in the body! Valid ONLYif θis in rad & ω is in rad/s!

  13. Sect. 10.2: Kinematic Equations • Recall: 1 dimensional kinematic equations for uniform (constant) acceleration (Ch. 2). • We’ve just seen analogies between linear & angular quantities: Displacement & Angular Displacement: x  θ Velocity & Angular Velocity: v  ω Acceleration & Angular Acceleration: a  α • For α= constant, we can use the same kinematic equations from Ch. 2 with these replacements!

  14. For α= constant, & using the replacements, x  θ, v  ω a  α we get these equations: NOTE:These are ONLY VALID if all angular quantities are in radian units!!

  15. Example 10.1: Rotating Wheel • A wheel rotates with constant angular accelerationα = 3.5 rad/s2. It’s angular speed at time t = 0 is ωi= 2.0 rad/s. (A) Find the angular displacement Δθit makes after t = 2 s. Use: Δθ = ωit + (½)αt2 = (2)(2) + (½)(3)(2)2 = 11.0 rad (630º) (B) Find the number of revolutions it makes in this time. Convert Δθfrom radians to revolutions: A full circle = 360º = 2πradians = 1 revolution 11.0 rad = 630º = 1.75 rev (C) Find the angular speed ωfafter t = 2 s. Use: ωf = ωi + αt = 2 + (3.5)(2) = 9 rad/s

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