In today’s article, we are going to study about ‘ Things you should learn about Coriolis force to score high ‘ How beneficial it is and what is coroilis Force. We will also read about some more topics like transformation in a rotating frame of reference (coroilis force), transformation of velocity (generalised form), transformation of acceleration etc in Detail. So, take your notebooks in you hand and get ready to study physics in an easy and sorted way.
Transformation in a rotating frame of reference (Coriolis Force)-
Coriolis effect is a deflection of moving object when the motion is described relating to a rotating frame of reference. In a reference frame with a clockwise direction, rotation. The deflection is to the left of motion. When reference frame rotates in anticlockwise direction then deflection is in the right side of motion.
Coriolis Force is proportional to the rotational state and it is act perpendicular to the rotational axis and the velocity of the body in the rotating frame.
Theory and derivation-
S is the stationary frame of reference and position vector of P w.r.t S is r= xi + yj + zk …(1)
Where r= OP
S’ is the rotating frame of reference which rotates with constant angular velocity then the position vector of point P is
r’ = x’i + y’j + z’k …(2)
For stable point P,
Put the value of r’ from (2),
r = x’i’ + y’j’ + z’k’ …(3)
Transformation of velocity (Generalised form)
Differentiate reaction (3) with respect to time t,
dr/dt = [i’ dx’/dt + j’ dy’/dt + k’ dz’/dt ] + [ x’ di’/dt + y’ dj’/dt + z’ dk’/dt ] …(4)
Velocity of particle with respect to S’ frame is given by:-
V’ = d’r’/dt = d’r/ dt = [ i’ dx’/dt + j’ dy’/dt + k’ dz’/dt ] …(5)
Here we replace d/dt by d’/dt and i’ , j’ , k’ are constant in S’ frame and r= r’.
Similarly the velocity of particle w.r.t S frame.
V= dr/dt = [ i dx/dt + j dr/dt + k dz/dt ] …(6)
Here i, j, and k are constant hence there differentiation are zero.
Now taking a vector R of a unit magnitude i.e. R= i+ j+ k moving with constant velocity ω then we can write
V= ω × R
dr/ dt = ω × R …(7)
We know i’, j’ and k’ are also moving with ω velocity so that with the help of equation (7) we can write-
di’/ dt = ω × i’
dj’/ dt = ω × j’
dk’/dt = ω × k’
Above these equations are listed as equation (8).
Now put the value from (5) and (8) into equation (4),
dr/dt = d’r/dt + (ω × i’x’ + ω × j’y’ + ω × k’z’)
dr/dt = d’r/dt + ω ( i’x’ + j’y’ + k’z’)
V= V’ + ω × r’
We know r’ = r
V = V’ + ω × r’
V= V’ + ω × r’
Also, V’ = V- ω × r’ …(9)
V’ = Velocity of particle with respect to S’ frame
V= Velocity of particle with respect to S frame
ω = Velocity of S’ frame
(2) Transformation of acceleration-
We have dr/dt = d’r/dt + ω × r
Rearranging the term,
d(r)/dt = (d’/dt + ω ×) r
From this equation we know value of operator on velocity,
a = d/dt (dr/dt) = (d’/dt + ω×) (dr/dt)
Now put the value of dr/dt on RHS
d/dt (dr/dt) = (d’/dt + ω×) (d’r/dt + ω × r)
a = d^2/dt^2 = d’/dt( d’r/dt) + d’/dt (ω × r) + ω × dr’/dt + ω ×(ω × r)
a = d^2/dt^2 = d’^2r/dt( d’r/dt) + d’/dt (ω × r’) + ω × dr’/dt + ω ×(ω × r)
a = d^2/dt^2 = d’^2r/dt^2 + d’ω/dt × r + ω × dr’/dt + ω ×(ω × r)
a= d’^2r/dt^2+ 2( ω × dr/dt) + ω ×(ω × r)
∴ ω = constant hend d’ω/dt = 0
a = a’ + 2(ω × v’) + ω ×(ω × r)
a’ = a – 2(ω × v’) + ω ×(ω × r)
This is transformation equation of acceleration,
a’ = acceleration of particle with respect to time s’ frame
a= acceleration of particle with respect to time s’ frame
(ω × v’)= Coriolis acceleration
ω ×(ω × r)= centrifugal acceleration
Note- Coriolis acceleration is due to the motion of particle in S’ frame.
Centrifugal acceleration is due to the motion of frame S’. Its direction will be opposite to the centripetal acceleration, (outwards).
So our article is finished and after completely reading this article, one can easily tell what is coroilis force. And we have also discussed some more topics transformation in a rotating frame of reference (coroilis force), transformation of velocity (generalised form), transformation of acceleration etc.
So one can say that they got a detailed information about coroilis force, and basic information about other topics.
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