# Mechanics in Noninertial Frames

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Mechanics in Noninertial Frames

Why Frames of Reference?
Newton’s Equations : are meaningless without frame of reference, w.r.t which, the position vector, and consequently the acceleration vector, are measured.

A reference frame is a rigid body with three reference directions attached to it
x y z

Are Newton’s Equations (Consequently whole of mechanics) valid in all kinds of reference frames? Answer : Newton’s equations are valid in a special class of frames known as the inertial frames.

Inertial Frame (Uncritical Definition) :
A frame that moves with constant velocity without rotation With respect to what ? Another Inertial Frame!!!

A more fundamental definition is provided by Newton’s First Law :
A frame in which a body, not acted upon by any force, is either at rest or moves uniformly Q. Does there exist such a frame? A. If there exists one such frame, then there exist a whole class of infinitely many such frames, as provided by Galelian Principle of Invariance

Galelian Principle of Invariance
If Newton’s equations of motion are valid in one frame, then they are valid in any other frame, moving with uniform velocity without rotation, w.r.t. it

Proof : S : Inertial Frame in which Newton’s Laws are valid
S’ : Another Frame, moving uniformly w.r.t S (force does not change)

Newton’s Grand Belief : There is one such frame, the absolute frame
Newton’s Grand Belief : There is one such frame, the absolute frame. The space, with respect to which this frame is at rest, is the absolute space A frame, that is very close to being an inertial frame, is the Celestial Sphere

Red Frame : Frame of the Celestial Sphere
Yellow Frame : Frame fixed to the Earth

Consequence : Earth is a little less close to being an inertial frame, as it slowly rotates with respect to the celestial sphere

Newton’s Equation in an accelerated frame
S : Inertial Frame S’ : Accelerated Frame

However, Where, and, Thus, Newton’s equations of motion are still valid in an accelerated frame, provided, a fictitious force is added to the real force

Fictitious forces (also called inertial force) on objects in an accelerated frame are proportional to their masses! Mpua0 Mpia0 Mpiaa0

Uniform Gravity! Mpug Mpig Mpiag Earth

The two forces, inertial and uniform gravity are totally indistinguishable
One can banish uniform gravity by accelerating his frame in an appropriate manner Earth g

One can take two viewpoints
1. Inertial forces are as real as gravity 2. Gravity is as fictitious as inertial forces Einstein took the second viewpoint and created general theory of relativity

One can make uniform gravity completely disappear by dropping his frame in this gravitational field
What about nonuniform gravity? Can it be also made to vanish? Ans : The major part of it can be made to vanish. However, a residual part will still remain, and this residual gravity is the well known Tidal Force

Tidal Force

Earth g0 Earth

Tidal Force of the Moon on Earth

r P1 P2 R

Distribution of Tidal Gravity on the surface of Earth
Moon Earth Distribution of Tidal Gravity on the surface of Earth

Low tide High tide Earth Moon

The Moon Wins Over the Sun in its Tidal Effects on Earth !!
Sun’s Gravity on Earth is Much Stronger than that of Moon : However, 27

Roche Limit Roche Limit

R a Equating the two :

If the tidal gravity of the primary over a secondary becomes stronger than the self gravity of the secondary, the secondaty is torn apart. Inside the Roche Limit, no object can be held together by its gravitational attraction alone

If the densities of the primary and its satellite are the same, then
A more realistic calculation, taking into account the deformation of the secondary before break up gives

Roche Limit for Saturn

For a comet, whose density is low, the Roche Limit is much larger
The comet Scoemaker Levy, Breaking up after entering the Roche limit of Jupiter

A truck at rest has one door fully open.
w Prob. 8.2 A truck at rest has one door fully open. The truck accelerates forward at constant rate A, and the door begins to swing shut. The door is uniform and solid, has total mass M, height h, and width w. a. Find the instantaneous ang. Velocity of the door about its hinges, when it has swung through 90 degrees. b. Find the horizontal force on the door when it has swung through 90 deg.

a) mA A Truck’s Frame Writing Integrating, Or,

b) Truck’s Frame F MA

Ground Frame F MA The net acceleration (w.r.t. the ground) of the CM of the door in the forward direction and in the position shown, is :

Prob. 3 The Schuler Pendulum
It is a pendulum, which is such that, it always points towards the centre of the earth, no matter what the motion of its point of suspension is, so long as the point of suspension moves in the local horizontal direction

Ordinary Pendulum (The Plumb line)
The plumb line defines the local vertical, so long as the point of suspension is at rest or is moving uniformly

The plumb line will no more follow the local vertical
mg ma The plumb line will no more follow the local vertical

Under the bouncy movement of Professor Calculus, the pendulum has a tough time finding the local vertical

Captain Haddock can be a more serious non-inertial frame!
Thundering typhoons! Why cannot your pendulum behave, Professor Calculus?

Prob. 3 Find the time period of the Schuler pendulum

S : Distance between point of suspension and CG
FFII a s FFI CG S : Distance between point of suspension and CG FFI & FFII : fictitious forces on the pendulum 46

If the pendulum has to always point towards the centre of the earth, it must rotate about P with an angular acceleration that must be the same as the ang. acc. of P about the centre of the earth. Now, the period of a compound pendulum is :

Plugging in the values :
we get : However, solving for s, we get If

Thus, Schuler pendulum is just a concept, however, a very useful concept in Inertial Navigation System

Rotating Coordinate System
x y z x’ y’ z’ (x,y,z) : Inertial frame (x’,y’,z’) : Frame rotating w.r.t the inertial frame Goal : Find the equation of motion of a particle in frame (x’,y’,z’) 50

Result I Change in a vector that undergoes infinitesimal rotation about a fixed axis : Unit vector along axis of rotation If the vector is rotating about the axis with angular velocity , then

Conversely, if the coordinate frame is rotating with angular velocity , a fixed vector will appear to rotate in the reverse sense. So,

Result II Relationship between rates of change of a vector in two frames, rotating w.r.t. each other z (x,y,z) : Inertial frame z’ y’ (x’,y’,z’) : Frame rotating w.r.t the inertial frame y x x’ Green arrow : A vector changing with time

Let : Rate of change of in the inertial frame Rate of change of in the rotating frame Q : How are & related to each other? Claim :

Proof : Let & be changes in the vector as observed in the two frames, in time . The two would be the same, if there were no relative rotation between the two However, due to the rotation of the primed frame, there will be an additional change in this frame This additional change is :

Velocity and Acceleration Vectors in the Two Frames
Substituting from the first equation,

Equation of Motion in the Rotating Frame
Multiplying both sides of the above eq. by the mass of the particle : However, Newton’s eq. being valid in an inertial frame,

Therefore, Eq. of motion in the rotating frame is :
Thus, eq. of motion of a particle in a rotating frame is in the form of Newton’s equation, provided, following fictitious forces are added : The Centrifugal Force The Coriolis Force Unnamed, as mostly absent

Centrifugal Force

Marble on a Roulette Wheel
Coriolis Force v Marble on a Roulette Wheel

On a merry-go-round in the night
Coriolis was shaken with fright Despite how he walked ‘Twas like he was stalked By some fiend always pushing him right David Morin, Eric Zaslow ……

Example I : Car on a Revolving Platform (Chapter 3)
Car driven along a fixed radial line with uniform velocity Equation of motion of the car in the frame of the platform :

Since in the platform’s frame the car is moving with uniform velocity,
64

Rotating Vessel of Water
Example II : Rotating Vessel of Water The surface of a fluid must follow a gravitational equipotential surface

Constant potential surface is given by :
(x,y,z) Constant potential surface is given by :

If R is the radius of the cylindrical vessel, then the depth of the surface is :

L : Length of massless rod : Const. Ang. Vel. of platform
mg T Prob : A pendulum is fixed on a revolving platform as shown. It can swing only in a plane perpendicular to the horizontal axle. M : Mass of pendulum L : Length of massless rod : Const. Ang. Vel. of platform Solution The forces on the pendulum are shown. The frame of reference is the rotating platform.

mg T

Coriolis Effect on Earth’s Surface
x y z This component alone, is mainly responsible for Coriolis effects The Coriolis force on a body moving on the surface of the earth is as if the earth is rotating with angular velocity about the vertical

Coriolis Effect on Cruise Missiles
(Taking ) For a cruise missile fired at a target distant L away, the deviation S is :

8.9. A 400 tons train runs south at a speed of 60 mi/h at a latitude of 60. What is the force on the tracks? What is the direction of the force? x y z The force, being to the right of direction of motion, is to the west (red arrow)

Foucault Pendulum

Foucault pendulum on a rotating platform
v x y The only force in the direction is the centrifugal force : Eq. of motion in the direction :

A solution for is : Thus, the plane of oscillations of the pendulum, rotates about the vertical with an angular velocity, which is the same as that of the rotating platform, but in the opposite sense.

Foucault Pendulum on Rotating Earth
On the surface of the earth, at latitude , the rate of rotation is : , where is earth’s angular velocity. The rotation is clockwise. x y z

At the north pole, the plane of the pendulum, rotates once a day.

Coriolis force and hurricane formation
Northern hemisphere Southern hemisphere

Low

Northern Hemisphere

Southern Hemisphere

High Pressure Hurricanes Do not Form

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