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

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?

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

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

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

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

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

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

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

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

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

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

Centrifugal Force

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