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Hour 26 Accelerating Reference Frames III
Physics 321 Hour 26 Accelerating Reference Frames III
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Forces on a Turntable 𝑚 𝑟 ′ = Effective force in S’
𝐹 𝑟𝑒𝑎𝑙 Actual forces (in S’ basis) +𝑚 𝑟 ′ × Ω ′ Transverse force +2𝑚 𝑟 ′ × Ω ′ Coriolis force +𝑚 Ω ′ × 𝑟 ′ × Ω ′ Centrifugal force
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Forces on the Earth 𝑚 𝑟 ′ = Effective force on the earth
𝐹 𝑟𝑒𝑎𝑙 Actual forces in earth basis +2𝑚 𝑟 ′ × Ω ′ Coriolis force +𝑚 Ω ′ × 𝑟 ′ × Ω ′ Centrifugal force
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The Earth x0, y0, z0 fixed in space x’, y’, z’ rotate with the earth
Polaris z0 x0, y0, z0 fixed in space x’, y’, z’ rotate with the earth Ω z’ points outward x’ points east y’ points north z’ y’ y0 x’ x0
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The Earth Ω =Ω cos λ 𝑦 +Ω sin λ 𝑧
Polaris z0 Ω =Ω cos λ 𝑦 +Ω sin λ 𝑧 Ω= 2𝜋 ×3600 𝑠 =7.29× 10 −5 𝑠 −1 Ω 𝑧 𝑦 λ z’ λ x0
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The Earth Ω =Ω cos λ 𝑦 +Ω sin λ 𝑧
Polaris z0 Ω =Ω cos λ 𝑦 +Ω sin λ 𝑧 𝑚 𝑟 ′ = 𝐹 𝑟𝑒𝑎𝑙 +𝑚 𝑟 ′ × Ω ′ +2𝑚 𝑟 ′ × Ω ′ +𝑚 Ω ′ × 𝑟 ′ × Ω ′ −𝑚 𝐴 ′ 𝑚 𝑟 =𝑚 𝑔 0 +2𝑚 𝑟 × Ω +𝑚 Ω × 𝑟 × Ω 𝑚 𝑟 =𝑚 𝑔 +2𝑚 𝑟 × Ω Ω 𝑧 𝑦 λ z’ λ x0
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The Earth Ω =Ω cos λ 𝑦 +Ω sin λ 𝑧 𝑚 𝑟 =𝑚 𝑔 +2𝑚 𝑟 × Ω
Polaris z0 Ω =Ω cos λ 𝑦 +Ω sin λ 𝑧 𝑚 𝑟 =𝑚 𝑔 +2𝑚 𝑟 × Ω 𝑚 𝑔 =𝑚 𝑔 0 +𝑚 Ω × 𝑟 × Ω =−𝑚 𝑔 0 𝑧 +𝑚 Ω 2 𝑟 cos λ 𝑢 Ω 𝑧 𝑦 λ 𝑢 z’ λ x0 𝑚 𝑔 =−𝑚 𝑔 0 𝑧 + 𝑚 Ω 2 𝑟 cos λ cos λ 𝑧 − sin λ 𝑦
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The Earth Greatly exaggerated: The direction of 𝑢 is 90° from Polaris.
z0 Ω 𝑦 𝑧 𝑢 𝑚 𝑔 x0 Greatly exaggerated: The direction of 𝑢 is 90° from Polaris. The surface of the earth is perpendicular to 𝑔 .
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Applications of the Coriolis Force
𝑚 𝑟 =𝑚 𝑔 +2𝑚 𝑟 × Ω Falling Stone Ascending Missile Air masses
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L Northern Hemisphere Low
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