Physics 319 Classical Mechanics G A Krafft Old

Physics 319 Classical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 17 Undergraduate Classical Mechanics Spring 2017

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? Undergraduate Classical Mechanics Spring 2017

Lagrangian Version • No energy conservation! Why? • Frame acceleration “causes” the inertial force Undergraduate Classical Mechanics Spring 2017

Pendulum in Accelerating Car • Using car-fixed coordinates • Question: what direction does a balloon go when you accelerate? Undergraduate Classical Mechanics Spring 2017

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 Undergraduate Classical Mechanics Spring 2017

Potential for Tidal Force • The potential function yielding the tidal force • Ocean surface a surface of constant potential for Undergraduate Classical Mechanics Spring 2017

Spring and Neap Tides • Taylor gives numbers hmoon = 54 cm, hsun = 25 cm • Spring tides effects from moon and sun add Sun • Neap tides effects at 90 degrees and “cancel” Sun Undergraduate Classical Mechanics Spring 2017

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 Undergraduate Classical Mechanics Spring 2017

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 Undergraduate Classical Mechanics Spring 2017

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 Undergraduate Classical Mechanics Spring 2017

Time Derivatives in a Rotating Frame • Suppose the prime frame rotates relative to an unprimed inertial frame with angular velocity Undergraduate Classical Mechanics Spring 2017

Acceleration in Rotating Frame • In rotating frame Newton’s second law is • 3 D centrifugal force • 3 D Coriolis force Undergraduate Classical Mechanics Spring 2017

Centrifugal Force Undergraduate Classical Mechanics Spring 2017

Plumb Line Shift • Tangential acceleration • Angle shift (south of vertical) Undergraduate Classical Mechanics Spring 2017