3.2 Torque, Moment Arms, and Exercise Sticking Points

4. Torque Mechanics and Moment Arms

In human movement, muscles do not produce linear displacement directly; they produce rotation of bony levers around joint axes. The quantitative measure of rotational force is Torque ($\tau$ or $T$), also called the moment of force.

The Torque Equation:

$\text{Torque }(\tau) = \text{Force }(F) \times \text{Moment Arm }(d_\perp)$

Where:

• $F$ = Magnitude of applied force (measured in Newtons, $\text{N}$).
• $d_\perp$ (Moment Arm) = The shortest perpendicular distance from the axis of rotation (joint center) to the line of action of the force (measured in meters, $\text{m}$).
• Torque ($\tau$) = Expressed in Newton-meters ($\text{N}\cdot\text{m}$) or foot-pounds ($\text{ft}\cdot\text{lb}$).

+-----------------------------------------------------------------------------------------+
|                           TORQUE & MOMENT ARM MECHANICS                                 |
|                                                                                         |
|                      Perpendicular Distance (d_perp)                                    |
|                      |<----------------------------->|                                  |
|                      |                               |                                  |
|                      |                               |                                  |
|        (Joint Axis)  ●===============================+                                  |
|         [Fulcrum]             (Forearm Bone)         |                                  |
|                                                      |                                  |
|                                                      ▼ Line of Action of External Load  |
|                                                     [Fr: Dumbbell]                      |
|                                                                                         |
|        External Torque (τ_ext) = Load Force (Fr) × Resistance Moment Arm (d_perp)       |
+-----------------------------------------------------------------------------------------+

Internal vs. External Torque

For any joint movement, two opposing torques interact:

1. Internal Muscle Torque ($\tau_{int}$): $\tau_{int} = \text{Muscle Contractile Force }(F_m) \times \text{Internal Moment Arm }(d_{\perp m})$ The internal moment arm is determined by anatomical insertion distance and joint angle.

2. External Resistance Torque ($\tau_{ext}$): $\tau_{ext} = \text{External Load Force }(F_r) \times \text{External Moment Arm }(d_{\perp r})$ The external moment arm is the horizontal distance from the joint axis to the vertical vector of gravitational pull on the weight.

Net Joint Acceleration Conditions:

• Concentric Acceleration: $\tau_{int} > \tau_{ext}$ (Muscle torque exceeds load torque; lever rotates in direction of muscle pull).
• Isometric Equilibrium: $\tau_{int} = \tau_{ext}$ (Torques are perfectly balanced; no joint rotation occurs).
• Eccentric Deceleration: $\tau_{ext} > \tau_{int}$ (Load torque exceeds muscle torque; muscle elongates under controlled tension).

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5. Mathematical Worked Example: Biceps Curl Torque Calculation

To understand the magnitude of internal muscular forces required due to mechanical disadvantage, consider the following clinical biomechanical scenario:

Scenario Parameters:
- Weight of dumbbell: 20 kg (Force = 20 kg × 9.81 m/s² = 196.2 N)
- Distance from elbow joint axis to dumbbell center of mass (dr): 0.35 m (35 cm)
- Forearm segment weight: Ignored for simplified calculation
- Distance from elbow joint axis to Biceps Brachii insertion (de): 0.04 m (4 cm)
- Elbow angle: 90 degrees (forearm parallel to the floor, where moment arms are maximized)

Step 1: Calculate External Resistance Torque ($\tau_{ext}$)

$\tau_{ext} = F_r \times d_r = 196.2\text{ N} \times 0.35\text{ m} = 68.67\text{ N}\cdot\text{m}$

Step 2: Calculate Required Muscle Effort Force ($F_m$) for Isometric Hold ($\tau_{int} = \tau_{ext}$)

$\tau_{int} = F_m \times d_e = 68.67\text{ N}\cdot\text{m}$ $F_m \times 0.04\text{ m} = 68.67\text{ N}\cdot\text{m}$ $F_m = \frac{68.67\text{ N}\cdot\text{m}}{0.04\text{ m}} = 1,716.75\text{ N}$

Biomechanical Takeaway:

To hold a 20 kg (~44 lb) dumbbell at $90^\circ$, the biceps brachii must exert $1,716.75\text{ N}$ of tension (equivalent to supporting ~175 kg or 385 lb)! This dramatic force requirement illustrates the immense internal joint loading created by third-class anatomical levers ($MA = 0.04 / 0.35 = 0.114$).

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6. Joint Angle Variations, Moment Arms, and Sticking Points

In free-weight resistance training, the line of gravitational resistance is strictly vertical. Consequently, as a limb rotates through its range of motion, the perpendicular moment arm ($d_\perp$) continuously changes:

• When the body segment is parallel to the floor (horizontal), the perpendicular distance to the vertical gravity line reaches its absolute maximum ($d_\perp = \text{maximum}$), producing peak external resistance torque.
• When the body segment is perpendicular to the floor (vertical / in line with gravity), the moment arm shrinks to zero ($d_\perp = 0$), producing zero external resistance torque.

+-----------------------------------------------------------------------------------------+
|                        MOMENT ARM VARIATION: DUMBBELL BICEPS CURL                       |
|                                                                                         |
|    [A] Start Position (180° / Full Ext)  [B] Mid-Range (90° Flexion)   [C] End (30° Flexion)   |
|                                                                                         |
|               O (Elbow Joint)                  O (Elbow Joint)               O          |
|               |                                |                             | /        |
|               | (dr ≈ 0)                       |=======[dr: MAX]======>      |/         |
|               |                                                       [Dumbbell]       |
|               ▼ [Dumbbell]                                                              |
|        External Torque: MINIMAL             External Torque: PEAK       External Torque: REDUCED|
|        (Weight in line with arm)            (Forearm parallel to floor) (Moment arm shortened)  |
+-----------------------------------------------------------------------------------------+

Biomechanical Sticking Points in Key Exercises:

1. Standing Dumbbell Biceps Curl:

   • At full elbow extension ($0^\circ$), the dumbbell hangs vertically below the elbow ($d_\perp \approx 0$).
   • At $90^\circ$ of flexion, the forearm is horizontal, creating the maximal resistance moment arm. This corresponds to the physiological sticking point where clients frequently fail.
   • Beyond $90^\circ$ (towards full flexion), the forearm rotates upward, shortening the moment arm and reducing external torque.

2. Standing Dumbbell Lateral Raise:

   • At the starting position (arms at sides), the dumbbell is vertically aligned with the glenohumeral joint ($d_\perp = 0$). External torque on the deltoid is near zero.
   • As the arm abducts to $90^\circ$ (parallel to the floor), the moment arm equals the entire arm length (~60-70 cm). Peak resistance torque occurs precisely at the top of the movement, making it the most difficult point.

3. Barbell Back Squat:

   • At the bottom of the squat (thighs parallel to the floor), the horizontal distance between the barbell (center of mass) and the hip and knee joint axes reaches its maximum.
   • The sticking point occurs just above parallel as the lifter attempts to reverse momentum while hip and knee extension moment arms are still near their peak.