Chapter 10 - Chapter 10: Rotations rigid body, stationary...

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Chapter 10: Rotations rigid body, stationary axis of rotation rotational variables (and their similarity to translations): • angle: θ • angular velocity: ϖ= d θ /dt • angular acceleration: α= d 2 θ /dt 2 = d ϖ /dt point at distance r from axis: relationship between v and ϖ rotations and kinetic energy moment of inertia
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Definition: rigid body Distance between any two points does not change A B A B
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Fixed axis of rotation For now, we consider only the case when the axis of rotation, wherever it is, is stationary
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Translation and Rotation Translation: all points move and have exactly the same displacements Rotation: all points rotate around some axis by exactly the same angle x-axis x 2 x 1 θ 2 θ 1 axis of rotation
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Translation and Rotation Translation coordinate: x (meters) Rotation angle: θ (radians) x-axis x 2 x 1 θ 2 θ 1 axis of rotation
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Radians DO NOT USE DEGREES (many formulas will work only in radians) s r θ s r θ = s = arc length 180º = π rad
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Equations of motion Translation coordinate: x(t) velocity: v = dx/dt acceleration: a = d 2 x/dt 2 Rotation angle: θ (t) angular velocity: ϖ ϖϖ = d θ /dt angular acc.: α αα = d 2 θ /dt 2 x-axis x 2 x 1 θ 2 θ 1 axis of rotation
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More equations of motion Translation Rotation 2 1 2 1 0 2 1 0 0 2 ( ) ( ) ( ) avg dx v t dt x x v t t a const v t v at x t x v t at = - = - = = + = + + 2 1 2 1 0 2 1 0 0 2 ( ) avg d t dt t t const t t t θ ϖ α = - = - = = + = + +
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Velocity and Angular Velocity s r θ ( ) ( ) ( ) ( ) ( ) ( ) s t t r s t r t s t r t v r θ ϖ = = = = v we already knew that!
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Acceleration and Angular Acceleration s r θ ( ) ( ) ( ) ( ) t v t t r v t t r a r ϖ α = = = a t 2 r v a r = a r v we already knew that! this is new!
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Period and Frequency 2 2 1 2 r T v f T π ϖ = = = = v we already knew that!
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Example 1
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Example 1
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A dancer spins in a pirouette movement at an initial angular velocity of ϖ =5 rad/s, then pulls in her leg and speeds up to ϖ =25 rad/s during 0.2s. What is her average
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Chapter 10 - Chapter 10: Rotations rigid body, stationary...

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