Hour 26 Accelerating Reference Frames III Physics 321 Hour 26 Accelerating Reference Frames III
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
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
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
The Earth Ω =Ω cos λ 𝑦 +Ω sin λ 𝑧 Polaris z0 Ω =Ω cos λ 𝑦 +Ω sin λ 𝑧 Ω= 2𝜋 23.9345×3600 𝑠 =7.29× 10 −5 𝑠 −1 Ω 𝑧 𝑦 λ z’ λ x0
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
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
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 .
Applications of the Coriolis Force 𝑚 𝑟 =𝑚 𝑔 +2𝑚 𝑟 × Ω Falling Stone Ascending Missile Air masses
L Northern Hemisphere Low