1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 17
G. A. Krafft
Jefferson Lab

2. Accelerating and Nonrotating Frames
Suppose one is in a frame that accelerates with respect to an inertial frame. Let the unprimed coordinates be those in an inertial frame, the primed coordinates be those in an accelerating frame, represent the position of the origin of the accelerating system in the inertial frame, and let the unit vectors of the two systems agree. Then
Suppose a vector force acts on a particle of mass m. What are the equations of motion in the accelerating frame?

3. Lagrangian Version No energy conservation! Why?
Frame acceleration “causes” the inertial force

4. Pendulum in Accelerating Car
Using car-fixed coordinates
Question: what direction does a balloon go when you accelerate?

5. Tides Two tides a day “proves” you are pulled to moon and sun!
Model: ocean of nearly uniform depth throughout earth. Mass in oceans small compared to total mass in earth

6. Potential for Tidal Force
The potential function yielding the tidal force
Ocean surface a surface of constant potential for

7. Spring and Neap Tides Taylor gives numbers hmoon = 54 cm, hsun = 25 cm
Spring tides effects from moon and sun add
Neap tides effects at 90 degrees and “cancel”
Sun
Sun
Sun
Sun

8. Rotation
Angular velocity vector: vector along the instantaneous rotation axis whose magnitude is the angular frequency ω
Velocity of a fixed position in the rotating frame
3 dimensional version of the plane rotation relation v = ωr
Magnitude and direction can depend on time

9. Rotation of Unit Vectors
Suppose a frame rotates with angular velocity How do the unit vectors fixed in the frame change in time?
Clearly consistent with general formula

10. Addition of Angular Velocities
Suppose frames 1, 2, and 3 have a common origin and, as viewed in frame 1 the rotation of frame 2 is , the rotation of frame 3 as viewed in frames 1 and 2 are
and respectively. Because relative velocities add
By the rotation velocity formula
for all ,
Angular velocities add as vectors

11. Time Derivatives in a Rotating Frame
Suppose the prime frame rotates relative to an unprimed inertial frame with angular velocity

12. Acceleration in Rotating Frame
In rotating frame Newton’s second law is
3 D centrifugal force
3 D Coriolis force

13. Centrifugal Force

14. Plumb Line Shift Tangential acceleration
Angle shift (south of vertical)