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Newton’s Laws in Rotational Language - Chapter 10 Summary

Explore the dynamics of rotating bodies through new angular concepts and Newton’s Laws in rotational motion. Learn about angular quantities like displacement, velocity, and acceleration. Understand kinematic equations for rotational motion.

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Newton’s Laws in Rotational Language - Chapter 10 Summary

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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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