2 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.
3 A reference frame is a rigid body with three reference directions attached to it xyz
4 Natural Question to ask : 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.
5 Inertial Frame (Uncritical Definition) : A frame that moves with constant velocity without rotationWith respect to what ?Another Inertial Frame!!!
6 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 uniformlyQ. 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
7 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
8 Proof : S : Inertial Frame in which Newton’s Laws are valid S’ : Another Frame, moving uniformly w.r.t S(force does not change)
9 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 spaceA frame, that is very close to being an inertial frame, is the Celestial Sphere
10 Red Frame : Frame of the Celestial Sphere Yellow Frame : Frame fixed to the Earth
11 Consequence : Earth is a little less close to being an inertial frame, as it slowly rotates with respect to the celestial sphere
12 Newton’s Equation in an accelerated frame S : Inertial FrameS’ : Accelerated Frame
13 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
14 Fictitious forces (also called inertial force) on objects in an accelerated frame are proportional to their masses!Mpua0Mpia0Mpiaa0
16 The two forces, inertial and uniform gravity are totally indistinguishable One can banish uniform gravity by accelerating his frame in an appropriate mannerEarthg
17 One can take two viewpoints 1. Inertial forces are as real as gravity2. Gravity is as fictitious as inertial forcesEinstein took the second viewpoint and created general theory of relativity
18 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
31 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
32 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
35 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
36 A truck at rest has one door fully open. wProb. 8.2A 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.
39 Ground FrameFMAThe net acceleration (w.r.t. the ground) of the CM of the door in the forward direction and in the position shown, is :
40 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
41 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
42 The plumb line will no more follow the local vertical mgmaThe plumb line will no more follow the local vertical
43 Under the bouncy movement of Professor Calculus, the pendulum has a tough time finding the local vertical
44 Captain Haddock can be a more serious non-inertial frame! Thundering typhoons! Why cannot your pendulum behave, Professor Calculus?
45 Prob. 3 Find the time period of the Schuler pendulum
46 S : Distance between point of suspension and CG FFIIasFFICGS : Distance between point of suspension and CGFFI & FFII : fictitious forces on the pendulum46
47 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 :
48 Plugging in the values : we get :However, solving for s, we getIf
49 Thus, Schuler pendulum is just a concept, however, a very useful concept in Inertial Navigation System
50 Rotating Coordinate System xyzx’y’z’(x,y,z) : Inertial frame(x’,y’,z’) : Frame rotating w.r.tthe inertial frameGoal : Find the equation of motion of a particle in frame (x’,y’,z’)50
51 Result IChange in a vector that undergoes infinitesimal rotation about a fixed axis: Unit vector along axis of rotationIf the vector is rotating about the axis with angular velocity , then
52 Conversely, if the coordinate frame is rotating with angular velocity , a fixed vector will appear to rotate in the reverse sense. So,
53 Result IIRelationship between rates of change of a vector in two frames, rotating w.r.t. each otherz(x,y,z) : Inertial framez’y’(x’,y’,z’) : Frame rotating w.r.tthe inertial frameyxx’Green arrow : A vector changing with time
54 Let :Rate of change of in the inertial frameRate of change of in the rotating frameQ : How are & related to each other?Claim :
55 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 twoHowever, due to the rotation of the primed frame, there will be an additional change in this frameThis additional change is :
57 Velocity and Acceleration Vectors in the Two Frames Substituting from the first equation,
58 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,
59 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 ForceThe Coriolis ForceUnnamed, as mostly absent
61 Marble on a Roulette Wheel Coriolis ForcevMarble on a Roulette Wheel
62 On a merry-go-round in the night Coriolis was shaken with frightDespite how he walked‘Twas like he was stalkedBy some fiend always pushing him rightDavid Morin, Eric Zaslow ……
63 Example I : Car on a Revolving Platform (Chapter 3) Car driven along a fixed radial line with uniform velocityEquation of motion of the car in the frame of the platform :
64 Since in the platform’s frame the car is moving with uniform velocity, 64
65 Rotating Vessel of Water Example II :Rotating Vessel of WaterThe surface of a fluid must follow a gravitational equipotential surface
66 Constant potential surface is given by : (x,y,z)Constant potential surface isgiven by :
67 If R is the radius of the cylindrical vessel, then the depth of the surface is :
68 L : Length of massless rod : Const. Ang. Vel. of platform mgTProb : 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 pendulumL : Length of massless rod: Const. Ang. Vel. of platformSolutionThe forces on the pendulum are shown. The frame of reference is the rotating platform.
70 Coriolis Effect on Earth’s Surface xyzThis component alone, is mainly responsible for Coriolis effectsThe 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
71 Coriolis Effect on Cruise Missiles (Taking )For a cruise missile fired at a target distant L away, the deviation S is :
72 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?xyzThe force, being to the right of direction of motion, is to the west (red arrow)
74 Foucault pendulum on a rotating platform vxyThe only force in the direction is the centrifugal force :Eq. of motion in the direction :
75 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.
76 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.xyz
77 At the north pole, the plane of the pendulum, rotates once a day.
78 Coriolis force and hurricane formation Northern hemisphereSouthern hemisphere