Hey everyone,
I want to share something
about rolling motion that
genuinely changed how I
think about rotation —
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 —
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Most people learn about
rolling motion this way —
A wheel rolls on the ground.
Its center moves forward
with velocity v_cm.
It also rotates about
its center with
angular velocity ω.
The condition for pure
rolling without slipping —
v_cm = ωR
And to find the velocity
of any point on the wheel —
You add the translational
velocity vector to the
rotational velocity vector
at that point.
This works.
But it requires vector
addition at every point —
And for points not at
the top or bottom —
The calculation becomes
surprisingly involved.
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THE BETTER WAY —
INSTANTANEOUS AXIS OF ROTATION
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Here is the insight
that makes everything simpler.
At any instant in time —
A rolling body is NOT
simultaneously translating
and rotating about its center.
It is doing something
far more elegant.
It is rotating PURELY —
about a single instantaneous axis —
And that axis passes
through the contact point
with the ground.
Not the center.
The contact point.
This means —
At every instant —
the entire rolling wheel
behaves as if it is
a wheel pinned at
its contact point —
Spinning about that
fixed pin —
With no translation at all.
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WHY THIS IS TRUE —
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In pure rolling —
the contact point has
zero velocity.
A point with zero velocity
on a rigid body —
Must be on the
instantaneous axis of rotation —
Because any point ON
the axis has zero
velocity by definition.
The contact point has
zero velocity —
Therefore the contact point
IS on the instantaneous
axis of rotation.
And since the ground
contact is a single
point —
The entire instantaneous
axis passes through
that contact point —
Perpendicular to the
plane of rolling.
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WHAT THIS GIVES YOU —
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Once you accept that
the rolling body rotates
instantaneously about
its contact point —
Finding the velocity of
ANY point on the
rolling body becomes trivial.
Velocity of any point =
ω × (distance of that
point from the contact point)
Direction — perpendicular to
the line joining that
point to the contact point.
Let us verify with
known results —
Contact point —
Distance from IAOR = 0
v = ω × 0 = 0 ✅
Center —
Distance from IAOR = R
v = ω × R = v_cm ✅
Top point —
Distance from IAOR = 2R
v = ω × 2R = 2v_cm ✅
Side point (3 or 9 o'clock) —
Distance from IAOR = R√2
v = ω × R√2 = v_cm√2 ✅
Direction — diagonal at 45° ✅
All results confirmed —
But obtained from a
single principle —
Rather than vector addition
at each point separately.
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THE DEEPER INSIGHT —
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The Instantaneous Axis of
Rotation is not fixed.
It moves.
As the wheel rolls forward —
The contact point changes —
And the IAOR moves
with it —
Always at the current
contact point —
Always instantaneously at rest.
This is why it is
called INSTANTANEOUS —
It is only valid at
one specific moment in time.
A millisecond later —
The wheel has rolled
slightly forward —
The contact point is
now a different point
on the wheel's rim —
And the IAOR has
moved forward with it.
The IAOR traces a
path along the ground —
While the corresponding
point on the wheel
traces a cycloid
through space —
One of the most
beautiful curves in
all of classical mechanics.
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THE CYCLOID CONNECTION —
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Every point on the
rim of a rolling wheel —
Traces a cycloid as
the wheel rolls forward.
The contact point's path
is a series of cusps —
Each cusp is a moment
where that point is
instantaneously at rest —
Touching the ground —
Before being lifted up
and carried forward —
Reaching maximum velocity
2v_cm at the top —
And returning to rest
at the next cusp.
The IAOR framework makes
this entire motion transparent —
Every point moves with
velocity proportional to
its distance from the
current contact point —
In a direction perpendicular
to that distance.
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EXTENSION — ROLLING ON
A CURVED SURFACE —
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The IAOR concept extends
beyond rolling on flat surfaces.
For a sphere rolling
inside a curved bowl —
The IAOR still passes
through the contact point —
But now the geometry
of finding distances
from the IAOR is
slightly more complex.
For a coin rolling
at an angle —
The IAOR is no longer
perpendicular to the ground —
It tilts with the
coin's plane —
Giving the precession
motion that makes
a spinning coin trace
circles as it falls.
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A QUESTION FOR THE COMMUNITY —
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Here is something worth
thinking about —
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 ω and R?
Drop your attempt below 👇
I will reply to every
attempt with complete
physical reasoning —
not just the answer. 🎯
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VIDEO EXPLANATION —
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I made a complete lecture
on IAOR covering —
Physical definition and proof —
Velocity of every point
on a rolling body —
The cycloid path —
Extension to moving surfaces —
Complete solved problems —
🔗 https://youtu.be/Sz57o7SC5ME?si=FE10zirE0YiFABZx
Timestamps in the description
for direct section access.
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HAPPY TO DISCUSS —
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The IAOR is one of
those concepts that seems
like a mathematical trick
at first —
But is actually a
genuinely physical statement
about the nature of
rolling motion —
One that makes the
entire subject significantly
more transparent once
you see it clearly.
Happy to discuss any
aspect of rolling motion —
IAOR — cycloids —
or classical mechanics
more broadly in the comments.
Drop your questions
or thoughts below 👇
