A Balanced Toe-Hang Putter
Introduction
In a previous post, I analyzed how different putter designs create torques that want to twist the face during your stroke. The key point was that putter head geometry creates two independent torque sources:
- Gravitational torque: This term is present even when the putter is held still. It depends on how far the center of mass is, in the heel-toe direction, from the shaft line when in the address position.
- Inertial torque: This term depends on how hard you accelerate the putter and on the distance from the shaft’s pivot point to the putter’s center of mass in the left-to-right direction (i.e. face to back of club).
Traditional blade style toe-hang putters fight the natural arc geometry by closing the face during the backswing since the inertial torque dominates. But what if we could design a toe-hang putter where these torques cancel out?
This post explores the “balanced toe-hang” design—a putter that achieves near-zero initial torques by deliberately using opposing gravitational and inertial forces.
The Cancellation Condition
For the total torque to be zero:
\[\tau_{total} = \tau_g + \tau_{in} = 0\]Substituting the torque formulas:
\[mgb_g + mab_a = 0\]The mass \(m\) cancels out, giving us the design constraint:
\[gb_g = -ab_a\]Rearranging for the critical ratio:
\[\frac{b_g}{b_a} = -\frac{a}{g}\]where:
- \(b_g\) = front/back COM offset from shaft (mm)
- \(b_a\) = heel/toe COM offset from shaft (mm)
- \(a\) = stroke acceleration (m/s²)
- \(g\) = gravitational acceleration (9.8 m/s²)
For example, a stroke acceleration of \(a = 9\) m/s² requires:
\[\frac{b_g}{b_a} = -\frac{9}{9.8} \approx -0.918\]Calibration
If we assume a relative shaft-to-COM length of \(b_a = -50\) mm, meaning the shaft is positioned to the heel side of the COM by about 2 inches:
\[b_g = -0.918 \times (-50) = 45.9 \text{ mm}\]Which just requires the shaft is also in front of the COM (toward the face) by about 2 inches also. This can be easily achieved through a large mallet with rear weight.
Comparison
| Parameter | Traditional Toe-Hang | Balanced Toe-Hang |
|---|---|---|
| \(b_g\) (forward offset) | +19 mm | +46 mm |
| \(b_a\) (heel/toe offset) | -51 mm | -50 mm |
| \(\tau_g\) @ rest | +0.065 N·m | +0.158 N·m |
| \(\tau_{in}\) @ 9 m/s² | -0.159 N·m | -0.158 N·m |
| Net torque | -0.094 N·m | ~0 N·m |
The balanced toe-hang achieves cancellation by having:
- Large gravitational torque (wants to open face)
- Equally large inertial torque (wants to close face)
- Perfect cancellation at initial acceleration
Performance
The cancellation is acceleration-dependent. Here’s how rotation varies with stroke aggressiveness:
| Stroke Type | Acceleration | Traditional Toe-Hang | Balanced Toe-Hang |
|---|---|---|---|
| Gentle | 6 m/s² | -2.5° (closes) | +3.1° (opens) |
| Medium | 9 m/s² | -5.7° (closes) | ±0.0° (neutral) |
| Aggressive | 12 m/s² | -8.8° (closes) | *-3.1° (closes) |
Observations:
- Traditional toe-hang: Always closes, fighting the required ~7° arc opening
- Balanced toe-hang: Neutral at 9 m/s², with ±3° variation across realistic stroke speeds
The balanced design provides ±3° variation compared to traditional toe-hang’s 6° range, but is still sensitive to tempo and rhythm (which manifest as acceleration) in the stroke.
When Two Wrongs Make a Right
Unlike zero torque putters that try to eliminate both offsets (\(b_g \approx 0\), \(b_a \approx 0\)), the balanced toe-hang uses large offsets in precise ratio to cancel their effects. Both torques are individually large (~0.16 N·m), but they oppose each other.
Putters exactly like these can be seen all over, too. I suspect this is more or less what the plumber’s neck version of the popular Taylor Made Spyder putters do. Especially with their changable weights, I imagine they can get those pretty dialed.
Even more players are reducing torque than we thought!