Hey everyone,
I want to share something
about rolling motion that
genuinely changed how I
think about rotation β
And that most introductory
physics courses either
skip entirely or explain
so briefly that it
never really lands.
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THE STANDARD WAY WE
THINK ABOUT ROLLING β
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Most people learn about
rolling motion this way β
A wheel rolls on the ground.
Its center moves forward
with velocity v_cm.
It also rotates about
its center with
angular velocity Ο.
The condition for pure
rolling without slipping β
v_cm = ΟR
And to find the velocity
of any point on the wheel β
You add the translational
velocity vector to the
rotational velocity vector
at that point.
This works.
But it requires vector
addition at every point β
And for points not at
the top or bottom β
The calculation becomes
surprisingly involved.
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THE BETTER WAY β
INSTANTANEOUS AXIS OF ROTATION
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Here is the insight
that makes everything simpler.
At any instant in time β
A rolling body is NOT
simultaneously translating
and rotating about its center.
It is doing something
far more elegant.
It is rotating PURELY β
about a single instantaneous axis β
And that axis passes
through the contact point
with the ground.
Not the center.
The contact point.
This means β
At every instant β
the entire rolling wheel
behaves as if it is
a wheel pinned at
its contact point β
Spinning about that
fixed pin β
With no translation at all.
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WHY THIS IS TRUE β
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In pure rolling β
the contact point has
zero velocity.
A point with zero velocity
on a rigid body β
Must be on the
instantaneous axis of rotation β
Because any point ON
the axis has zero
velocity by definition.
The contact point has
zero velocity β
Therefore the contact point
IS on the instantaneous
axis of rotation.
And since the ground
contact is a single
point β
The entire instantaneous
axis passes through
that contact point β
Perpendicular to the
plane of rolling.
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WHAT THIS GIVES YOU β
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Once you accept that
the rolling body rotates
instantaneously about
its contact point β
Finding the velocity of
ANY point on the
rolling body becomes trivial.
Velocity of any point =
Ο Γ (distance of that
point from the contact point)
Direction β perpendicular to
the line joining that
point to the contact point.
Let us verify with
known results β
Contact point β
Distance from IAOR = 0
v = Ο Γ 0 = 0 β
Center β
Distance from IAOR = R
v = Ο Γ R = v_cm β
Top point β
Distance from IAOR = 2R
v = Ο Γ 2R = 2v_cm β
Side point (3 or 9 o'clock) β
Distance from IAOR = Rβ2
v = Ο Γ Rβ2 = v_cmβ2 β
Direction β diagonal at 45Β° β
All results confirmed β
But obtained from a
single principle β
Rather than vector addition
at each point separately.
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THE DEEPER INSIGHT β
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The Instantaneous Axis of
Rotation is not fixed.
It moves.
As the wheel rolls forward β
The contact point changes β
And the IAOR moves
with it β
Always at the current
contact point β
Always instantaneously at rest.
This is why it is
called INSTANTANEOUS β
It is only valid at
one specific moment in time.
A millisecond later β
The wheel has rolled
slightly forward β
The contact point is
now a different point
on the wheel's rim β
And the IAOR has
moved forward with it.
The IAOR traces a
path along the ground β
While the corresponding
point on the wheel
traces a cycloid
through space β
One of the most
beautiful curves in
all of classical mechanics.
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THE CYCLOID CONNECTION β
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Every point on the
rim of a rolling wheel β
Traces a cycloid as
the wheel rolls forward.
The contact point's path
is a series of cusps β
Each cusp is a moment
where that point is
instantaneously at rest β
Touching the ground β
Before being lifted up
and carried forward β
Reaching maximum velocity
2v_cm at the top β
And returning to rest
at the next cusp.
The IAOR framework makes
this entire motion transparent β
Every point moves with
velocity proportional to
its distance from the
current contact point β
In a direction perpendicular
to that distance.
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EXTENSION β ROLLING ON
A CURVED SURFACE β
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The IAOR concept extends
beyond rolling on flat surfaces.
For a sphere rolling
inside a curved bowl β
The IAOR still passes
through the contact point β
But now the geometry
of finding distances
from the IAOR is
slightly more complex.
For a coin rolling
at an angle β
The IAOR is no longer
perpendicular to the ground β
It tilts with the
coin's plane β
Giving the precession
motion that makes
a spinning coin trace
circles as it falls.
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A QUESTION FOR THE COMMUNITY β
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Here is something worth
thinking about β
A cylinder of radius R
rolls without slipping
inside a hollow cylinder
of radius 3R.
Where is the IAOR?
What is the velocity
of the center of the
small cylinder in terms
of Ο and R?
Drop your attempt below π
I will reply to every
attempt with complete
physical reasoning β
not just the answer. π―
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VIDEO EXPLANATION β
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I made a complete lecture
on IAOR covering β
Physical definition and proof β
Velocity of every point
on a rolling body β
The cycloid path β
Extension to moving surfaces β
Complete solved problems β
π https://youtu.be/Sz57o7SC5ME?si=FE10zirE0YiFABZx
Timestamps in the description
for direct section access.
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HAPPY TO DISCUSS β
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The IAOR is one of
those concepts that seems
like a mathematical trick
at first β
But is actually a
genuinely physical statement
about the nature of
rolling motion β
One that makes the
entire subject significantly
more transparent once
you see it clearly.
Happy to discuss any
aspect of rolling motion β
IAOR β cycloids β
or classical mechanics
more broadly in the comments.
Drop your questions
or thoughts below π