PHY 6200 Theoretical Mechanics Chapter 9 Motion in a non-inertial reference frame Prof. Claude A Pruneau Notes compiled by L. Tarini

Introduction Sometimes it is simply easier to write equations in a non-inertial reference frame. –E.g. Motion near the surface of the Earth A non inertial reference frame

Rotating coordinate systems Rotating system Fixed system

First consider a fixed point in the rotating frame

Next, consider a moving point in the rotating frame Motion of the point in the rotating frame: Motion of the point in the fixed frame:

Consider in rotating system Assume fixed and rotating system have same origin in the rotating system

Example: Consider a d   rotation about the 3-axis.

Similarly consider a d   rotation about the 2-axis.

Similarly Conclusion

The result is valid for any vector. Note that, the angular acceleration, is the same in both systems.

Now define Conclusion

Acceleration is valid only in inertial frames

The last term

Finally… For an observer in the rotating coordinate system Sum of the forces acting on the particle as measured in the fixed (inertial) frame Translation + angular acceleration Centrifugal Force Coriolis Force

Centrifugal and Coriolis forces are not forces in the usual sense of the word. They are pseudo-forces introduced by our desire to write (non inertial terms) We thus have:

Directed outward The Centrifugal Term Magnitude: Direction:

Example 2 Hockey puck on a large merry-go-round (M-G-R) with a smooth frictionless horizontal flat surface Assume M-G-R has constant angular velocity rotating clockwise (seen from above). a) Find the effective force on the hockey puck after it has been given a push. b) Plot path

Solution: Neglecting friction; measured by observer on rotating surface. Assume the puck is initially at Initial velocity

Motion relative to the Earth In the fixed inertial frame External forces Gravitational attraction along a radius Assume the Earth is a perfect sphere and neglect the fact varies because of oblateness, density, nonuniformities, altitude changes, etc.

Practically constant in time So we neglect

Since near the Earth, the centrifugal force is dominated by The centrifugal force is responsible for the oblate shape of the Earth.

Earth is deformed Equatorial radius is km > polar radius at the pole relative to the equator On a calm ocean, the water is perpendicular to, not

Effective (Pendulum period + direction of equilibrium) about 0.35% of In simpler terms

Coriolis Force Term Magnitude Direction north Deflected path to the right of the particle motion

Basics Properties of the Jovian Planets PlanetDistance (AU) Period (years) Diameter (km) Mass (Earth=1) Density (g/cm3) Rotation (hours) Jupiter Saturn Uranus Neptun e Jet stream flows eastward Jupiter : speed ~ 300 km/h Saturn : speed ~ 1300 km/h Light zones – upwelling air,capped by white ammonia cirrus clouds – top of enormous convection current. Darker zones – cooler atmosphere – downward motion – complete convection cycle.

Example: Find horizontal deflection of a plumb lines caused by Coriolis effect for a particle falling from height “h” Solution: Initially…

Ex 5 Pendulum Precession (Foucault pendulum) Small oscillations z x y

z x y

Let That’s a damped oscillator! Except we have an imaginary term. The solution is thus:

if then oscillation frequency thus rotation (precession) with