Transcript Physics 321 Hour 26 Accelerating Reference Frames III

Physics 321
Hour 26
Accelerating Reference Frames III
Forces on a Turntable
𝑚𝑟′ =
𝐹𝑟𝑒𝑎𝑙
Effective force in S’
Actual forces (in S’ basis)
+𝑚𝑟 ′ × Ω′
+2𝑚𝑟 ′ × Ω′
+𝑚 Ω′ × 𝑟 ′ × Ω′
Transverse force
Coriolis force
Centrifugal force
Forces on the Earth
𝑚𝑟′ =
𝐹𝑟𝑒𝑎𝑙
+2𝑚𝑟 ′ × Ω′
+𝑚 Ω′ × 𝑟 ′ × Ω′
Effective force on the earth
Actual forces in earth basis
Coriolis force
Centrifugal force
The Earth
Polaris
z0
x0, y0, z0 fixed in space
x’, y’, z’ rotate with the earth
Ω
y’
x’
z’
z’ points outward
x’ points east
y’ points north
y0
x0
The Earth
Polaris
z0
Ω
𝑦
z’
λ
Ω = Ω cos λ 𝑦 + Ω sin λ 𝑧
𝑧
λ
2𝜋
Ω=
23.9345 × 3600 𝑠
= 7.29 × 10−5 𝑠 −1
x0
The Earth
Polaris
z0
Ω
𝑦
𝑧
Ω = Ω cos λ 𝑦 + Ω sin λ 𝑧
λ
𝑚𝑟′
z’
λ
= 𝐹𝑟𝑒𝑎𝑙 + 𝑚𝑟 ′ × Ω′ + 2𝑚𝑟 ′
× Ω′ + 𝑚 Ω′ × 𝑟 ′ × Ω′
x0
− 𝑚𝐴′
𝑚𝑟
= 𝑚𝑔0 + 2𝑚𝑟 × Ω
+𝑚 Ω×𝑟 ×Ω
𝑚𝑟 = 𝑚𝑔 + 2𝑚𝑟 × Ω
The Earth
Polaris
z0
Ω
𝑦
Ω = Ω cos λ 𝑦 + Ω sin λ 𝑧
𝑧
λ
𝑢
z’
λ
𝑚𝑟 = 𝑚𝑔 + 2𝑚𝑟 × Ω
𝑚𝑔 = 𝑚𝑔0 + 𝑚 Ω × 𝑟 × Ω
= −𝑚𝑔0 𝑧 + 𝑚Ω2 𝑟 cos λ𝑢
x0
𝑚𝑔 = −𝑚𝑔0 𝑧 +
𝑚Ω2 𝑟 cos λ cos λ 𝑧 − sin λ 𝑦
The Earth
Polaris
z0
Ω
𝑦
𝑧
𝑢
𝑚𝑔
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
𝑚𝑟 = 𝑚𝑔 + 2𝑚𝑟 × Ω
• Falling Stone
• Ascending Missile
• Air masses
Northern Hemisphere Low
L