Transcript Physics 321 Hour 24 Accelerating Reference Frames I

Physics 321
Hour 24
Accelerating Reference Frames I
Bottom Line
• In linear systems, we have to add a term to
Newton’s 2nd Law to account for the
acceleration of the frame.
𝑚𝑟 = 𝐹 − 𝑚𝐴
𝑟 is the perceived acceleration
𝐹 is the true force
𝐴 is the acceleration of the frame
• In rotating frames, we can replace time
derivatives with 𝜔 ×… expressions.
Consider an accelerating train car
𝑡
𝑣𝑑𝑡
𝑥(𝑡)
0
𝑥0 (𝑡)
𝑣(𝑡)
Examples
• Glass of water in a car
• Tides
Two Angular Momentum Theorems
• Euler’s Theorem
Any tiny rotation can be considered as a
rotation about a fixed axis.
(Complex rotations can be taken to be a series of
infinitesimal rotations.)
• Tiny rotations commute.
(Larger rotations do not commute.)
Angular Velocity
• Angular velocity vector (r.h.r.) 𝜔 = 𝜔𝑢
• Let 𝑒 be a unit vector fixed in a body
𝑑𝑒
=𝜔×𝑒
𝑑𝑡
• In particular
𝑑𝑟
𝑑 𝑒𝑟
𝑣=
=𝑟
=𝜔×𝑟
𝑑𝑡
𝑑𝑡
• In general
𝑑𝑄
𝑑𝑄
=
+ Ω×𝑄 𝑆
𝑑𝑡 𝑆0
𝑑𝑡 𝑆
Transforming Velocity
𝑟0 = 𝑟 + Ω × 𝑟
𝑆