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Lecture18 - Moment of Inertia Discrete object Continuous...

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I = x 2 dm 0 M Moment of Inertia I = Σ i m i r i 2 Discrete object I = r 2 dm Continuous object r is the distance from the axis of rotation m 2 r 1 m 1 r 2 I = m 1 r 2 1 + m 2 r 2 2 dm x dm x dm = M/L dx K tot = K CM + K rot = 1/2 m v 2 + 1/2 I ω 2 Kinetic Energy Remember : I and K depend from the axis of rotation!
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Example : Which object will arrive first? h θ Sliding (box) : Potential Rolling : Potential I sphere (CM) = 2/5 M R 2 I cylinder (CM) = 1/2 M R 2 BALLS (Rolling) BOX (Sliding) Bodies will slide faster than they roll! Same M Kinetic translation Rotation Kinetic translation Faster?!
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h θ I sphere (CM) = 2/5 M R 2 I cylinder (CM) = 1/2 M R 2 BALLS (Rolling) Same M V = 2gh / β +1
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h θ
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Example : Which object will make it? Same M m R Y
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I sphere (CM) = 2/5 M R 0 2 I hoop (CM) = M R 0 2 I solid cylinder (CM) = 1/2 M R 0 2
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Example : A uniform thin rod of length L and mass M is pivoted at one end. Held horizontal and released from rest. The pivot is frictionless. Find: a) Angular velocity of the rod when it reaches its vertical position.
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