3.2 Torque, Moment Arms, and Exercise Sticking Points

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1 Chapter 1: Exam Overview

1.1 NCSF-CPT Exam Structure, Logistics, and Scoring

2 Chapter 2: Functional Anatomy: Structures and Movement

2.1 Basic Skeletal Anatomy and Bone Function 2.2 Bone Remodeling, Joint Classification, and Connective Tissues 2.3 Anatomical Position, Planes, Axes, and Joint Actions 2.4 Upper-Body Muscles and Functional Roles 2.5 Core, Spinal, and Lower-Body Muscles

3 Chapter 3: Functional Anatomy: Biomechanics and Kinetic Chain

3.1 Lever Classes and Mechanical Advantage 3.2 Torque, Moment Arms, and Exercise Sticking Points 3.3 Muscle Actions, Length-Tension, and Force-Velocity 3.4 Proprioception, Posture, Balance, and Kinetic-Chain Imbalances

4 Chapter 4: Exercise Physiology

4.1 ATP Turnover and the Phosphagen System 4.2 Glycolysis, Lactate, and Metabolic Byproducts 4.3 Oxidative Metabolism and the Energy-System Continuum 4.4 Motor Units and Excitation-Contraction Coupling 4.5 Muscle Fibers, Recruitment, and Proprioception 4.6 Acute Cardiorespiratory and Hemodynamic Responses 4.7 Thresholds, Chronic Adaptation, Warm-Up, and Detraining

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6 Chapter 6: Screening and Health Evaluation

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9 Chapter 9: Weight Management

9.1 Energy Balance and Components of Daily Expenditure 9.2 Resting Metabolism and TDEE Estimation 9.3 Exercise Energy Expenditure and Weight-Management Programming 9.4 Body-Composition Targets and Safe Weight Loss 9.5 Lean-Mass Preservation, Gain, and Metabolic Adaptation 9.6 Safe Weight Gain, Disordered Eating, and Common Myths

10 Chapter 10: Resistance Training Program Design

10.1 Overload, Progression, and Regression 10.2 Specificity, Variation, and Individualization 10.3 Reversibility, Recovery, and Maintenance 10.4 Exercise Selection and Order 10.5 Intensity, Volume, and Rest Intervals 10.6 Tempo, Frequency, and Session Structure 10.7 Programming Endurance, Hypertrophy, Strength, and Power 10.8 Periodization, Fatigue, and Deload Decisions

11 Chapter 11: Cardiorespiratory and Flexibility Programming

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12 Chapter 12: Training Instruction

12.1 Session Readiness, Acute Response, and Injury Prevention 12.2 Aerobic and Anaerobic Exercise Instruction 12.3 Speed, Power, Agility, Balance, and Coordination Instruction 12.4 Equipment Selection, Setup, and Safe Operation 12.5 Squat Setup and Execution 12.6 Squat Faults and Front-Squat Modifications 12.7 Deadlift, Lunge, and Step-Up Technique 12.8 Horizontal and Vertical Pressing Technique 12.9 Rowing and Pulldown Technique 12.10 Joint Stabilization and Core Exercise Instruction 12.11 Flexibility Instruction and Injury-Responsive Modification 12.12 Spotting, Liftoffs, and Pre-Lift Communication 12.13 Bailouts, Coaching Cues, and Motor Learning

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3.2 Torque, Moment Arms, and Exercise Sticking Points

Key Takeaways Explain This Topic AI

Torque equals force multiplied by the perpendicular moment arm to the axis.

Moving a load farther from a joint usually increases external torque at that joint.

A sticking region reflects changing leverage, muscle force capacity, technique, and acceleration rather than one universal angle.

Machines, cables, bands, and free weights create different resistance profiles because their force directions and moment arms change differently.

Last updated: August 2026

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 ( τ \tau τ or T T T ) , also called the moment of force.

The Torque Equation: Torque ( τ ) = Force ( F ) × Moment Arm ( d ⊥ ) \text{Torque }(\tau) = \text{Force }(F) \times \text{Moment Arm }(d_\perp) Torque ( τ ) = Force ( F ) × Moment Arm ( d ⊥ ​ )

Where:

F F F = Magnitude of applied force (measured in Newtons, N \text{N} N ).

d ⊥ d_\perp d ⊥ ​ (Moment Arm) = The shortest perpendicular distance from the axis of rotation (joint center) to the line of action of the force (measured in meters, m \text{m} m ).

Torque ( τ \tau τ ) = Expressed in Newton-meters ( N ⋅ m \text{N}\cdot\text{m} N ⋅ m ) or foot-pounds ( ft ⋅ lb \text{ft}\cdot\text{lb} ft ⋅ 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:

Internal Muscle Torque ( τ i n t \tau_{int} τ in t ​ ): τ i n t = Muscle Contractile Force ( F m ) × Internal Moment Arm ( d ⊥ m ) \tau_{int} = \text{Muscle Contractile Force }(F_m) \times \text{Internal Moment Arm }(d_{\perp m}) τ in t ​ = Muscle Contractile Force ( F m ​ ) × Internal Moment Arm ( d ⊥ m ​ )

The internal moment arm is determined by anatomical insertion distance and joint angle.

External Resistance Torque ( τ e x t \tau_{ext} τ e x t ​ ): τ e x t = External Load Force ( F r ) × External Moment Arm ( d ⊥ r ) \tau_{ext} = \text{External Load Force }(F_r) \times \text{External Moment Arm }(d_{\perp r}) τ e x t ​ = External Load Force ( F r ​ ) × External Moment Arm ( d ⊥ 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: τ i n t > τ e x t \tau_{int} > \tau_{ext} τ in t ​ > τ e x t ​ (Muscle torque exceeds load torque; lever rotates in direction of muscle pull).

Isometric Equilibrium: τ i n t = τ e x t \tau_{int} = \tau_{ext} τ in t ​ = τ e x t ​ (Torques are perfectly balanced; no joint rotation occurs).

Eccentric Deceleration: τ e x t > τ i n t \tau_{ext} > \tau_{int} τ e x t ​ > τ in t ​ (Load torque exceeds muscle torque; muscle elongates under controlled tension).

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 ( τ e x t \tau_{ext} τ e x t ​ ) τ e x t = F r × d r = 196.2 N × 0.35 m = 68.67 N ⋅ m \tau_{ext} = F_r \times d_r = 196.2\text{ N} \times 0.35\text{ m} = 68.67\text{ N}\cdot\text{m} τ e x t ​ = F r ​ × d r ​ = 196.2 N × 0.35 m = 68.67 N ⋅ m

Step 2: Calculate Required Muscle Effort Force ( F m F_m F m ​ ) for Isometric Hold ( τ i n t = τ e x t \tau_{int} = \tau_{ext} τ in t ​ = τ e x t ​ ) τ i n t = F m × d e = 68.67 N ⋅ m \tau_{int} = F_m \times d_e = 68.67\text{ N}\cdot\text{m} τ in t ​ = F m ​ × d e ​ = 68.67 N ⋅ m F m × 0.04 m = 68.67 N ⋅ m F_m \times 0.04\text{ m} = 68.67\text{ N}\cdot\text{m} F m ​ × 0.04 m = 68.67 N ⋅ m F m = 68.67 N ⋅ m 0.04 m = 1 , 716.75 N F_m = \frac{68.67\text{ N}\cdot\text{m}}{0.04\text{ m}} = 1,716.75\text{ N} F m ​ = 0.04 m 68.67 N ⋅ m ​ = 1 , 716.75 N

Biomechanical Takeaway:

To hold a 20 kg (~44 lb) dumbbell at 90 ∘ 90^\circ 9 0 ∘ , the biceps brachii must exert 1 , 716.75 N 1,716.75\text{ N} 1 , 716.75 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 ( M A = 0.04 / 0.35 = 0.114 MA = 0.04 / 0.35 = 0.114 M A = 0.04/0.35 = 0.114 ).

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 ⊥ d_\perp d ⊥ ​ ) 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 ⊥ = maximum d_\perp = \text{maximum} d ⊥ ​ = 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 ⊥ = 0 d_\perp = 0 d ⊥ ​ = 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:

Standing Dumbbell Biceps Curl:

At full elbow extension ( 0 ∘ 0^\circ 0 ∘ ), the dumbbell hangs vertically below the elbow ( d ⊥ ≈ 0 d_\perp \approx 0 d ⊥ ​ ≈ 0 ).

At 90 ∘ 90^\circ 9 0 ∘ 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 ∘ 90^\circ 9 0 ∘ (towards full flexion), the forearm rotates upward, shortening the moment arm and reducing external torque.

Standing Dumbbell Lateral Raise:

At the starting position (arms at sides), the dumbbell is vertically aligned with the glenohumeral joint ( d ⊥ = 0 d_\perp = 0 d ⊥ ​ = 0 ). External torque on the deltoid is near zero.

As the arm abducts to 90 ∘ 90^\circ 9 0 ∘ (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.

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.