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Navi_mansoori_2[2]_PPT_M.SC.1ST_SEM[1].pptx
- 1. APPLICATION OF CORIOLIS FORCE BY MOHD NAVI MANSOORI M.Sc. First Semester Enroll. No. 220/22 Mentor (Ass. Prof.) Dr. Devendra Singh Department of Physics School of Physical and Decision Science Baba saheb Bhimrao Ambedkar University ( A Central University) Lucknow-226025.UP
- 2. CONTENTS CORIOLIS FORCE CENTRIFUGAL FORCE CALCULATION OF CORIOLIS FORCE AND CENTRIFUGAL FORCE CORIOLIS FORCE EFFECT CORIOLIS EFFECT ( ON EARTH ) APPLICATION
- 3. CORIOLIS FORCE • Gaspard-Gustave de Coriolis, (born May 21, 1792, Paris—died September 19, 1843, Paris), French engineer and mathematician who first described the Coriolis force, an effect of motion on a rotating body .
- 4. DEFINITION “ Coriolis Force is an fictitious force(pseudo force) that acts on objects that are in motion relative to rotating reference frame. Deflection of an object due to the Coriolis force is called the Coriolis effect. It is caused by rotation of rotating body.
- 5. Calculation of Coriolis force and centrifugal force Let us consider two sets of coordinate axis, one of them is fixed (inertial frame of reference) as S(x,y,z) and other is rotating (non inertial frame of reference) as S’(x’, y’,z’) rotating with angular velocity 𝜔 about an axis passes through common origin O. The position vector of a particle at point P with respect to frame S is given as ; 𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘……………………………..(1) The position vector of same particle at point P with respect to S’ frame is given as ; 𝑟 = 𝑥′𝑖′ + 𝑦′𝑗′ + 𝑧′𝑘′ ……………………...(*2) Differentiate both side of equation with respect to ‘t’ : ⅆ𝑟 ⅆ𝑡 = 𝑖′ ⅆ𝑥′ ⅆ𝑡 + 𝑥′ ⅆ𝑖′ ⅆ𝑡 + 𝑗′ ⅆ𝑦′ ⅆ𝑡 + 𝑦′ ⅆ𝑗′ ⅆ𝑡 + 𝑘′ ⅆ𝑧′ ⅆ𝑡 + 𝑧′ ⅆ𝑘′ ⅆ𝑡
- 6. CONTINUE ⅆ𝑟 ⅆ𝑡 = ⅆ𝑥 ⅆ𝑡 𝑖′ + ⅆ𝑦′ ⅆ𝑡 𝐽′ + ⅆ𝑧′ ⅆ𝑡 𝑘′ + 𝑥′ ⅆ𝑖′ ⅆ𝑡 + 𝑦′ ⅆ𝐽′ ⅆ𝑡 + 𝑧′ ⅆ𝑘′ ⅆ𝑡 𝑉 𝑠 = 𝑉 𝑠 ′ + (𝑥′ ⅆ𝑖′ ⅆ𝑡 + 𝑦′ ⅆ𝐽′ ⅆ𝑡 + 𝑧′ ⅆ𝑘′ ⅆ𝑡 ) …………………………..(3) Where, 𝑉 𝑠 = velocity of particle w.r.t S And, 𝑉 𝑠 ′ = velocity of particle w.r.t. S’ As we know that: 𝑣 = 𝜔 × 𝑟 So, ⅆ𝑟 ⅆ𝑡 = 𝜔 × 𝑟 ………………………………………………………….(4) Replace r by unit vector in equation (4): ⅆ 𝑖′+𝑗′+𝑘′ ⅆ𝑡 = 𝜔 × 𝑖′ + 𝑗′ + 𝑘′
- 7. So, ⅆ𝑖′ ⅆ𝑡 + ⅆ𝑗′ ⅆ𝑡 + ⅆ𝑘′ ⅆ𝑡 = 𝜔 × 𝑖′ + 𝜔 × 𝑗′ + 𝜔 × 𝑘′ ………………………(5) Put equation (5) in equation (3) 𝑉 𝑠 = 𝑉 𝑠 ′ + 𝑥′𝜔 × 𝑖′ + 𝑦′𝜔 × 𝐽′ + 𝑧′𝜔 × 𝑘′ 𝑉 𝑠 = 𝑉 𝑠 ′ + 𝜔 × 𝑥′𝑖′ + 𝑦′𝑗′ + 𝑧′𝑘′ By using equation (2): 𝑉 𝑠 = 𝑉 𝑠 ′ + 𝜔 × 𝑟 ………………………………………………………………(6) ⅆ𝑟 ⅆ𝑡 𝑠 = ⅆ𝑟 ⅆ𝑡 𝑠′ + 𝜔 × 𝑟………………………………………….(7) Equation (7) is true for any other similar vector quantities, So replace r by 𝑉 𝑠 in (7) We have ; ⅆ𝑉𝑠 ⅆ𝑡 𝑠 = ⅆ𝑉𝑠 ⅆ𝑡 𝑠′ + 𝜔 × 𝑉 𝑠 ………………………..(8) Using (6) in R.H.S. of equation (8), ⅆ𝑉𝑠 ⅆ𝑡 𝑠 = ⅆ ⅆ𝑡 𝑉𝑠′ + 𝜔 × 𝑟 𝑠′ + 𝜔 × 𝑉𝑠′ + 𝜔 × 𝑟
- 8. ⅆ𝑉 𝑠 ⅆ𝑡 = ⅆ𝑉𝑠′ ⅆ𝑡 + ⅆ𝜔 ⅆ𝑡 × 𝑟 𝑠′ + 𝜔 × ⅆ𝑟 ⅆ𝑡 𝑠1 + 𝜔 × 𝑉𝑠′ + 𝑤 × 𝑤 × 𝑟 𝑎𝑠 = 𝑎𝑠′ + ⅆ𝜔 ⅆ𝑡 × 𝑟 + 𝜔 × 𝑉𝑠′ + 𝜔 × 𝑉𝑠′ + 𝜔 × 𝑤 × 𝑟 As ⅆ𝑟 ⅆ𝑡 𝑠′ = 𝑉𝑠′ 𝑎𝑠 = 𝑎𝑠′ + ⅆ𝑤 ⅆ𝑡 × 𝑟 𝑠′ + 2𝜔 × 𝑣𝑠′ + 𝜔 × 𝜔 × 𝑟 ……………….(9) Multiplying both sides by ‘m’ 𝑚𝑎𝑠 = 𝑚𝑎𝑠′ + 𝑚 ⅆ𝑤 ⅆ𝑡 × 𝑟 𝑠′ + 2𝑚(𝜔 × 𝑣𝑠′) + 𝑚𝜔 × 𝜔 × 𝑟 𝐹𝑠 = 𝐹𝑠′ + 𝑚 ⅆ𝑤 ⅆ𝑡 × 𝑟 𝑠′ + 2𝑚(𝜔 × 𝑣𝑠′) + 𝑚𝜔 × 𝜔 × 𝑟 𝐹𝑠′ = 𝐹𝑠 − [𝑚 ⅆ𝑤 ⅆ𝑡 × 𝑟 𝑠′ + 2𝑚(𝜔 × 𝑣𝑠′) + 𝑚𝜔 × 𝜔 × 𝑟 ] ………………………….(10)
- 9. 𝐹𝑠′ = 𝐹𝑠 − [𝑚 ⅆ𝑤 ⅆ𝑡 × 𝑟 𝑠′ + 2𝑚(𝜔 × 𝑣𝑠′) + 𝑚𝜔 × 𝜔 × 𝑟 ] 𝐹𝑠′ = 𝐹𝑠 + 𝐹0 Where 𝐹0 = Fictious force = −[𝑚 ⅆ𝑤 ⅆ𝑡 × 𝑟 𝑠′ + 2𝑚(𝜔 × 𝑣𝑠′) + 𝑚𝜔 × 𝜔 × 𝑟 ] If 𝜔 = constant, then ⅆ𝑤 ⅆ𝑡 = 0 So 𝐹0= fictious force = −[0 + 2𝑚(𝜔 × 𝑣𝑠′) + 𝑚𝜔 × 𝜔 × 𝑟 ] Fictious force = Coriolis force + centrifugal force Where Coriolis force = -𝟐𝒎(𝝎 × 𝒗𝒔′) And centrifugal force = -𝒎𝝎 × 𝝎 × 𝒓
- 10. CORIOLIS FORCE (EFFECTS) It is a apparent force . Due to rotation of the coordinate system ( Earth) . It makes a moving object deflect from a straight line even in the absence of any forces acting on it .
- 11. CORIOLIS EFFECT ON EARTH • The Coriolis effect, which is a result of Earth's rotation, causes moving particles such a air to be deflected to the right in the Northern Hemisphere and to the left in Southern Hemisphere.
- 12. APPLICATION It is taken into account to compute accurately trajectories of long range projectiles and missile. It is C.F. that produces counterclockwise circulation in northern hemishpere which effect course of winds. The spinning motion of the earth is that which causes the equatotial bulge.