Transcript Physics 321 Hour 25 Accelerating Reference Frames II

Physics 321
Hour 25
Accelerating Reference Frames II
Consider an accelerating train car
𝑡
𝑣𝑑𝑡
𝑥(𝑡)
0
𝑣(𝑡)
𝑥0 (𝑡)
𝑡
𝑥 𝑡 = 𝑥0 𝑡 −
𝑣(𝑡)𝑑 𝑡
0
𝑦 𝑡 = 𝑦0 (𝑡)
𝑥 𝑡 = 𝑥0 𝑡 − 𝑣(𝑡)
𝑥 𝑡 = 𝑥0 𝑡 − 𝑎(𝑡)
Proof
𝑑 𝑒𝑟
= 𝜔 × 𝑒𝑟
𝑑𝑡
∆𝑒𝑟 = 1 ∙ ∆𝜃
∆𝑒𝑟 ∆𝜃
∆𝑒𝑟
=
=𝜔
𝑒𝑟 (𝑡 = 0)
∆𝑡
∆𝑡
∆𝜃
𝑑𝑒𝑟
= 𝜔 × 𝑒𝑟
𝑑𝑡
Proof
𝑒𝜃 (𝑡 = 0)
∆𝑒𝜃
𝑑 𝑒𝜃
= 𝜔 × 𝑒𝜃
𝑑𝑡
∆𝑒𝜃 = 1 ∙ ∆𝜃
∆𝑒𝜃 ∆𝜃
∆𝜃
=
=𝜔
∆𝑡
∆𝑡
𝑑 𝑒𝜃
= 𝜔 × 𝑒𝜃
𝑑𝑡
Proof
𝑑𝑒𝑧
= 𝜔 × 𝑒𝑧 = 0
𝑑𝑡
Any unit vector can be written as
𝑒 = 𝑎𝑒𝑟 + 𝑏𝑒𝜃 + 𝑐 𝑒𝑧
In general
𝑑𝑄
𝑑𝑄
=
+ Ω×𝑄 𝑆
𝑑𝑡 𝑆0
𝑑𝑡 𝑆
where S0 is the lab (inertial) frame and S is a
frame rotating with angular velocity Ω.
Examples
Handout
rotation.nb