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Hey guys, Today, we'll talk about acceleration in rotating
reference frames (in the language of engineers, of course)
Basically, there are three kinds of acceleration. Acceleration could be radial, tangential,
or coriolis. Coriolis acceleration is a weird one, based on disagreement between reference
frames, whereas the radial and tangential components are responsible for the conic path
followed by the object. Now, let's see what these components represent.
Say, you're spinning a stone (with the help of a string) in the clockwise direction with
an angular velocity omega. In such a case, the radial acceleration is pointing inwards,
towards the driver (which is your hand). Because, you're pulling an object that's trying to
go straight, by which it traces the circular path.
As you speed up your spinning, tangential acceleration comes into play, which is nothing
but the tangent to the circle at the stone's position, directed to the right (as our motion
is clockwise here). The radial acceleration is nothing but a fancy
name for the centripetal acceleration - V^2/r. It's responsible for keeping the stone in
the circular path. But, this tangential acceleration a=r alpha, comes only when there's a change
in how fast you spin. Because this alpha is the angular acceleration, which is nothing
but the time derivative of angular velocity. Now, this could be zero, when you're either
at rest, or spinning at some constant omega. This directly influences the tangential component.
It becomes zero. Mind you, the radial component is still there. Again, that keeps the stone
in its circle. Now, to the weird force -- The Coriolis force.
Question, What happens when you throw a ball while you're spinning around? This happens.
Okay, let's say that we're riding a Merry-Go-Round, and I'm shooting balls from one end, while
you're at the other end trying to catch them. Will you ever catch them? Nope. The path of
the ball curves. You'll definitely miss the ball, unless I throw it very fast. Because,
you'd have moved sideways between the small interval of time. As a matter of fact, the
turbulence in our Earth's atmosphere is caused by the very same effect (due to the rotation
of the Earth of course). This is the Coriolis effect. You can google for more.
That's enough physics. Let's take a driving crank OA with a clockwise angular velocity.
It has a slider B attached to it. This slider is constrained only to move along the crank
(up & down). Assume that the slider has a velocity in the upward direction.
This looks very analogous to our ball-throwing experiment. Slider is the ball, and crank
is the Merry Go Round. The direction of this force is important because it resembles the
tangential acceleration of the crank (by the direction, at least). We'll talk about that
in acceleration diagrams. Anyways, there are four cases the direction
of Coriolis force. But, one case is enough to analyze the others. If the velocity of
B is upward, and OA rotates clockwise, then the Coriolis acceleration acts to the right.
It's a cross product (coriolis component acts normal to the plane containing omega and v)
and so, we follow the right hand rule. If the index finger gives the direction of velocity,
and the thumb represents the direction of angular velocity, then the middle finger is
the direction of Coriolis force. This means that the direction of coriolis
component changes when omega or v change (but not both). If both change, we get the same
direction again. You might be wondering how angular velocity
points in a direction. It's rotational, right? How shall we represent that with our thumb?
Well, we have a convention for that, given by the right palm rule. If the fingers point
in the direction of the curl of omega, then the thumb indicates our convention. Yeah,
it's up when omega is anticlockwise (dunno who chose that way) and down, when its clockwise.
Thank you.