This document explores Assertion #2:
Sway is used to control and dampen the effects of the Force of Inertia.
Our goal is to examine this assertion rigorously and explore the physical principles behind sway in swing dances (Waltz, Foxtrot, Quickstep).
Inertia is the resistance of a mass to changes in motion.
This includes:
Linear acceleration (change in speed/direction)
Rotational acceleration (change in angular velocity)
While inertial forces are not “real” in Newtonian physics (they’re pseudo-forces), they describe the effects of motion on mass within a rotating or accelerating frame.
In ballroom dance, inertial effects appear during:
| Situation | Description | Apparent Force Direction | Dancer's Reaction |
|-----------|-------------|---------------------------|--------------------|
| Linear Acceleration | Sudden changes in velocity | Opposite direction of change | Engage core and maintain poise |
| Turning (Angular Acceleration) | Body rotates during figure | Outward from turn center (centrifugal) | Apply sway inward to stabilize |
| Rise and Fall | Vertical body elevation | Up/down momentum affects stability | Adjust through ankle/knee/hip |
| Free Leg Swing | Limb mass adds lateral torque | Torque around spine/hip | Adjust spine or ribcage (sway) to compensate |
Sway is a mechanical response to inertia. It:
Repositions the spine to shift the Center of Mass (CoM)
Counters rotational forces during turning
Stabilizes the body during rise and swing
Brings CoG back over the foot
Absorbs lateral and rotational momentum
Prevents imbalance or “overturning”
To prove the assertion, we must demonstrate:
Inertial forces arise during turn and elevation.
Without sway, CoG would leave the base of support (BOS), destabilizing the dancer.
With sway, CoG is realigned over BOS, restoring balance.
The amount of sway correlates with the inertial load (more turn or speed = more sway).
We can simulate this using a simple physical model:
Two-mass pendulum (torso + leg)
Apply angular acceleration
Compare CoG paths with and without spinal inclination
Sway is not merely stylistic — it’s biomechanical.
It mitigates rotational and linear inertia.
Sway helps maintain dynamic equilibrium in figures with:
Turn
Rise/Fall
Swinging limbs
We will continue building on this to formalize sway's role as a stability mechanism in Swing dances.
“Sway is the elegant solution of the human body to a Newtonian inconvenience.”
– The Great Philosopher Nandhra
In figures like the Forward and Backward Locks in Quickstep, sway is used to manage linear inertia. As the dancer rises and then "falls" forward, potential energy (PE) is converted to kinetic energy (KE), generating high-speed travel.
Though there is minimal turn, sway appears subtly to:
Re-align the spine over the foot during rapid directional changes
Counteract torque from swinging limbs
Maintain balance during momentum-heavy travel
This form of sway is often subconscious and highly trained, but biomechanically critical to fluid, balanced movement in high-velocity figures.
“We need math to prove why we are right, not just say what someone else was told years ago by someone they can’t remember.”
– The Great Philosopher Nandhra
What We're Proving:
Sway is a biomechanical response to inertial forces — necessary to preserve balance during motion involving:
Rise/Fall
Acceleration (linear or angular)
Rotational inertia
To do this, we’ll show:
That inertial force shifts the effective CoG outside the BOS (base of support)
That without sway, this leads to instability
That adding a sway angle repositions the CoG back over the foot
That the sway angle correlates with inertial force magnitude (i.e., speed, mass, and angular change)
“We need math to prove why we are right, not just say what someone else was told years ago by someone they can’t remember.”
— The Great Philosopher Nandhra
To mathematically prove that:
Sway is a biomechanical necessity used to counteract inertial forces during turn and/or rise in ballroom dance.
Let:
\(m\): total dancer mass
\(m_b\): mass of upper body contributing to sway
\(g\): gravitational acceleration \(\approx 9.81 \text{ m/s}^2\)
\(h\): height of CoM from foot contact
\(v\): velocity of dancer's travel
\(r\): effective radius of turn
\(\omega\): angular velocity, \(\omega = \frac{v}{r}\)
\(\theta\): sway angle (angle from vertical)
When the dancer is turning at velocity \(v\), the apparent inertial force felt outward (from the rotating frame) is:
\[ F_c = \frac{mv^2}{r}\]
This acts laterally on the CoM, creating a torque:
\[ \tau_{\text{inertia}} = F_c \cdot h = \frac{mv^2}{r} \cdot h\]
Sway shifts the upper body’s center of mass sideways by:
\[ x = h \cdot \sin(\theta)\]
This shift creates a restoring torque:
\[ \tau_{\text{sway}} = m_b \cdot g \cdot x = m_b \cdot g \cdot h \cdot \sin(\theta)\]
For the dancer to maintain equilibrium:
\[ \tau_{\text{inertia}} = \tau_{\text{sway}}\]
\[ \frac{mv^2}{r} \cdot h = m_b \cdot g \cdot h \cdot \sin(\theta)\]
Cancel \(h\):
\[ \frac{mv^2}{r} = m_b \cdot g \cdot \sin(\theta)\]
Solve for \(\theta\):
\[ \sin(\theta) = \frac{mv^2}{r \cdot m_b \cdot g}\]
\[ \boxed{\theta = \sin^{-1} \left( \frac{mv^2}{r \cdot m_b \cdot g} \right)}\]
Faster movement (\(v \uparrow\)) ⇒ more sway required
Tighter turn (\(r \downarrow\)) ⇒ more sway required
More upper body control (\(m_b \uparrow\)) ⇒ less sway needed to produce same torque
Taller dancers (\(h \uparrow\)) have higher torque leverage
This equation proves that sway is not optional — it’s dictated by physics when movement and elevation are involved.
Sway is a biomechanical response to rotational and linear inertial forces.
Its magnitude can be derived from physical first principles.
It is not just styling — it is torque control.