Centre of Mass, Torque and Levers

Overview

This topic explores the fundamental mechanical principles that govern balance, stability, and the effectiveness of force application in human movement. Centre of mass and base of support determine whether an athlete remains balanced or falls. Levers explain how the musculoskeletal system amplifies force or speed, and torque quantifies the rotational effect of forces. These concepts are essential for understanding technique in virtually every sport — from maintaining balance on a beam to generating power in a golf swing to optimizing body position for a sprint start.

Centre of Mass (COM)

Definition

The centre of mass (also called centre of gravity when considering gravitational effects) is the point at which the entire mass of a body or system can be considered to be concentrated. It is the balance point of an object — the single point where the weighted position of all the mass is located.

Alternative Terms

• Centre of gravity (COG): The point where gravitational force effectively acts (same as COM in uniform gravitational field)

• Centre of mass (COM): The balance point based on mass distribution

• In sport biomechanics, these terms are often used interchangeably

Properties of Centre of Mass

1. Represents the entire body: For analysis purposes, the body can be treated as a point mass located at the COM

2. Follows parabolic path: During projectile motion, the COM follows a predictable parabolic trajectory regardless of body movements

3. Can be outside the body: For hollow or curved objects (like a bent body), the COM may be located in empty space

4. Changes with body position: As limbs move, the COM location shifts

5. Determines stability: The relationship between COM and base of support determines balance

Location of COM in the Human Body

Standing Position (Anatomical Position)

• Vertical location: Approximately at the level of the second sacral vertebra (S2)

• Height: Approximately 55-57% of total height from the ground

• Anterior-posterior: Slightly anterior to the spine

• Lateral: On the midline (in symmetrical stance)

Variations by Body Type and Sex

Approximate COM Height

COM Location Changes with Body Position

The COM shifts when body segments move:

COM Outside the Body

In certain body positions, the COM can be located in empty space:

Fosbury Flop High Jump:

        ___
       /   \
      |  *  |  ← COM (outside body)
       \___/
      
    =========== Bar
    
     \       /
      \_____/
       Body arched over bar

Examples where COM is outside the body:

• High jumper (Fosbury Flop): COM passes under or at bar level while body goes over

• Pike position in diving/gymnastics

• Hollow body position

• Back arch positions

Calculating Centre of Mass

For a System of Particles

$x_{COM} = \frac{\sum m_i x_i}{\sum m_i} = \frac{m_1x_1 + m_2x_2 + ... + m_nx_n}{m_1 + m_2 + ... + m_n}$

$y_{COM} = \frac{\sum m_i y_i}{\sum m_i}$

Where:

• $x_{COM}$, $y_{COM}$ = coordinates of centre of mass

• $m_i$ = mass of each particle/segment

• $x_i$, $y_i$ = coordinates of each particle/segment's centre of mass

Segmental Method for Human Body

The human body is modeled as linked segments, each with known:

• Mass (as percentage of total body mass)

• Centre of mass location (as percentage of segment length)

Body Segment Mass Percentages (approximate)

Worked Example: Calculating COM

A simplified example with three segments on the x-axis:

$x_{COM} = \frac{(5 \times 0.3) + (35 \times 0.5) + (30 \times 0.8)}{5 + 35 + 30}$

$x_{COM} = \frac{1.5 + 17.5 + 24}{70} = \frac{43}{70} = 0.614 \text{ m}$

COM in Projectile Motion

Key Principle: Once airborne, the COM follows a predictable parabolic path that cannot be changed (no external horizontal forces, constant vertical acceleration due to gravity).

However, athletes can:

• Move body parts relative to the COM

• Appear to change trajectory by repositioning segments

• Reach higher or further by strategic body positioning

Example — Long Jump:

• COM follows fixed parabola after takeoff

• "Hitch-kick" technique cycles legs to prepare for landing

• Arms and legs move, but COM path is unchanged

• Landing position optimizes distance measurement

Example — High Jump:

• Fosbury Flop allows body to arch over bar

• COM may pass at or below bar height

• Each body part clears bar sequentially

• Result: Clearing higher bars than COM height alone would allow

Stability and Balance

Definition

• Balance: The ability to maintain equilibrium or a desired body position

• Stability: The resistance to disruption of equilibrium; the ability to return to equilibrium after being disturbed

Static vs Dynamic Balance

Equilibrium

An object is in equilibrium when:

1. Sum of all forces = 0 (no linear acceleration)

2. Sum of all torques = 0 (no angular acceleration)

Types of Equilibrium

Base of Support (BOS)

Definition

The base of support is the area enclosed by the outermost edges of the body in contact with the supporting surface. It includes the area between the contact points.

Determining Base of Support

BOS includes:

• Area directly under feet/hands in contact with ground

• Area between contact points

• Any area where support could be provided if needed

Standing with feet apart:

    Left foot     Right foot
    ┌────┐       ┌────┐
    │    │       │    │
    │    │       │    │
    └────┘       └────┘
    
    ←───── Base of Support ─────→
    (includes area between feet)

Examples of Different Bases of Support

Factors Affecting BOS

Factors Affecting Stability

1. Position of Centre of Mass Relative to Base of Support

Principle: For stability, the line of gravity (vertical line through COM) must fall within the base of support.

Stable:                    Unstable:
   
   COM                        COM
    ●                          ●
    |                            \
    |                             \
    ↓                              ↓
┌────────┐                    ┌────────┐
│  BOS   │                    │  BOS   │ ← Line of gravity
└────────┘                    └────────┘   outside BOS
    
Line of gravity              About to fall!
within BOS

Key Points:

• Centre of BOS = most stable position

• Near edge of BOS = less stable

• Outside BOS = falling (must correct or fall)

• Athletes deliberately move COM near BOS edge to initiate movement

2. Height of Centre of Mass

Principle: Lower COM = greater stability

Explanation:

• Lower COM requires greater angular displacement to move outside BOS

• Lower COM = lower potential energy = more work required to tip over

• Lower COM = greater torque required to destabilize

Examples:

Sport Applications:

• Sumo wrestling: Low, wide stance

• Rugby scrum: Low body position

• Judo: Bend knees when being attacked

• Surfing/skateboarding: Lower stance in challenging conditions

3. Size of Base of Support

Principle: Larger BOS = greater stability

Explanation:

• Larger BOS means COM can move more before leaving the base

• More room for adjustment and correction

• Greater tolerance for external perturbations

Examples:

4. Mass of the Body

Principle: Greater mass = greater stability (if other factors equal)

Explanation:

• Greater mass = greater inertia = more force required to accelerate

• Greater mass = more momentum to overcome

• Greater mass = greater weight pressing down on surface (friction)

Example: A 120 kg rugby player is harder to push over than a 70 kg player (assuming similar positions)

5. Friction Between Body and Surface

Principle: Greater friction = greater stability

Explanation:

• Higher friction prevents sliding

• Allows athlete to push against ground to resist forces

• Low friction surfaces (ice, wet floor) reduce stability

Summary Table: Increasing Stability

Stability Equation (Qualitative)

$\text{Stability} \propto \frac{\text{BOS size} \times \text{Mass} \times \text{Friction}}{\text{COM height}}$

Stability in Sport Applications

Maximizing Stability (Defensive)

Minimizing Stability (Initiating Movement)

Dynamic Stability

In many sports, stability is dynamic — the athlete is constantly adjusting:

Examples:

• Surfing: Continuous COM adjustments over small BOS

• Gymnastics beam: Constant micro-corrections

• Ice hockey: Gliding while changing direction

• Skateboarding: Balance over moving platform

Anticipatory Postural Adjustments

Athletes often pre-adjust their posture before a perturbation:

• Weight shift before catching heavy object

• Leaning before being hit in contact sports

• Widening stance before collision

Levers

Definition

A lever is a simple machine consisting of a rigid bar (lever arm) that rotates about a fixed point (fulcrum) when a force (effort) is applied to overcome a resistance (load).

Components of a Lever

Visual Representation

           Effort (E)
              ↓
              │
    ┌─────────┼─────────────────┐
    │         │                 │
    │    EA   │       RA        │
    └─────────┴─────────────────┘
              △                  ↓
           Fulcrum            Load (R)

Lever Classes

Levers are classified based on the relative positions of the fulcrum (F), effort (E), and resistance/load (R).

First-Class Lever

Arrangement

Effort — Fulcrum — Resistance (F in the middle)

    E                    R
    ↓                    ↓
    │                    │
    └────────△───────────┘
          Fulcrum

Characteristics

• Fulcrum is between effort and resistance

• Can favor force OR speed, depending on arm lengths

• Changes direction of force application

Mechanical Advantage

• If EA > RA: Force advantage (like a crowbar)

• If EA < RA: Speed/distance advantage

• If EA = RA: No mechanical advantage (balanced)

Examples in Daily Life

• Seesaw/teeter-totter

• Scissors

• Crowbar

• Pliers

• Balance scale

Examples in the Human Body

1. Head/Neck (Nodding)

• Fulcrum: Atlanto-occipital joint (where skull meets spine)

• Effort: Neck extensor muscles (posterior)

• Resistance: Weight of face/head (anterior)

• Type: First-class (effort and resistance on opposite sides of fulcrum)

            ┌──────────────┐
   E (neck  │              │ R (face
   extensors)│     F       │  weight)
            └──────────────┘
                   △
              Atlanto-occipital
                  joint

2. Elbow Extension (Triceps) When pushing down against resistance with arm extended:

• Fulcrum: Elbow joint

• Effort: Triceps pulling on olecranon

• Resistance: Weight in hand or force against palm

Note: This is first-class when the forearm is being extended against resistance below the elbow.

Second-Class Lever

Arrangement

Effort — Resistance — Fulcrum (R in the middle)

    E              R
    ↓              ↓
    │              │
    └──────────────△
              Fulcrum

Characteristics

• Resistance is between effort and fulcrum

• Always provides force advantage (mechanical advantage > 1)

• Effort arm is always longer than resistance arm

• Sacrifices speed/range of motion for force

• Effort and resistance move in the same direction

Mechanical Advantage

• Always > 1 (force multiplier)

• Less effort required to move load

• Trade-off: Effort must move greater distance than load

Examples in Daily Life

• Wheelbarrow

• Nutcracker

• Bottle opener

• Door (hinge at fulcrum, push at edge)

• Nail clipper (lower lever)

Examples in the Human Body

1. Calf Raise / Plantarflexion (Primary Example)

• Fulcrum: Ball of foot (metatarsal heads)

• Resistance: Body weight at ankle joint (tibia)

• Effort: Gastrocnemius/soleus pulling on calcaneus (heel)

    ←── Foot ──→
    
    E (Achilles)    R (body weight)    F
         ↓               ↓              △
         │               │              │
    ┌────────────────────────────────────┐
    │         Calcaneus         Ball of │
    └────────────────────────────────────┘
         Heel                        Toe

Why Second-Class:

• Fulcrum is at toe

• Body weight acts through ankle (middle)

• Calf muscles pull up on heel (far from fulcrum)

• This arrangement allows powerful push-off despite heavy body weight

2. Pushing a Wheelbarrow (Human Using External Lever)

• Fulcrum: Wheel

• Resistance: Load in barrow

• Effort: Hands lifting handles

Third-Class Lever

Arrangement

Resistance — Effort — Fulcrum (E in the middle)

                   R              E
                   ↓              ↓
                   │              │
    △──────────────┴──────────────┘
 Fulcrum

Characteristics

• Effort is between fulcrum and resistance

• Always provides speed/distance advantage (mechanical advantage < 1)

• Resistance arm is always longer than effort arm

• Sacrifices force for speed and range of motion

• Effort and resistance move in the same direction

• Most common lever type in the human body

Mechanical Advantage

• Always < 1 (force disadvantage)

• Requires greater effort than resistance

• Trade-off: Resistance moves faster and further than effort

Examples in Daily Life

• Fishing rod

• Baseball bat

• Golf club

• Tweezers

• Broom

• Shovel (lower hand as fulcrum)

• Tennis racket

Examples in the Human Body

1. Elbow Flexion (Biceps Curl) — Classic Example

• Fulcrum: Elbow joint (humeroulnar articulation)

• Effort: Biceps brachii inserting on radial tuberosity (~3-5 cm from elbow)

• Resistance: Weight in hand (30-40 cm from elbow)

                Weight
                  ↓
    ┌─────────────●
    │           Hand
    │ Forearm
    │
    ○──────● ←── Biceps attachment
    │ Elbow
    │ (Fulcrum)

Calculation Example:

• Biceps inserts 4 cm from elbow

• Weight held 35 cm from elbow

• Mechanical advantage = 4/35 = 0.11

This means the biceps must generate ~9× the force of the weight being held!

Why Third-Class is Common in the Body: The body prioritizes speed and range of motion over force:

• Small muscle contraction → large limb movement

• Quick movements for sports and daily activities

• Muscles can contract with high force but limited distance

• Lever system amplifies movement distance

2. Knee Extension (Quadriceps)

• Fulcrum: Knee joint

• Effort: Quadriceps via patellar tendon (inserts on tibial tuberosity)

• Resistance: Weight of lower leg + any load at ankle/foot

3. Hip Flexion

• Fulcrum: Hip joint

• Effort: Hip flexors (iliopsoas) attaching to femur

• Resistance: Weight of leg

4. Shoulder Movements

• Most shoulder movements operate as third-class levers

• Deltoid raising arm against resistance

• Rotator cuff muscles in various movements

Summary of Lever Classes

Visual Summary

FIRST CLASS:    E ─────△───── R     (F in middle)
                      
SECOND CLASS:   E ─────R─────△      (R in middle)
                      
THIRD CLASS:    △─────E───── R      (E in middle)

Mechanical Advantage (MA)

Definition

Mechanical advantage is the factor by which a machine (lever) multiplies the input force. It is the ratio of the output force to the input force, or equivalently, the ratio of the effort arm to the resistance arm.

Formula

$MA = \frac{\text{Effort Arm (EA)}}{\text{Resistance Arm (RA)}}$

Or, at equilibrium: $MA = \frac{\text{Resistance Force}}{\text{Effort Force}} = \frac{R}{E}$

Interpretation

Calculating MA in the Human Body

Example: Biceps Curl

• Biceps inserts 5 cm from elbow

• Weight held 35 cm from elbow

$MA = \frac{EA}{RA} = \frac{5}{35} = 0.14$

Interpretation:

• MA < 1, so this is a speed/distance advantage system

• Biceps must produce ~7× the force of the weight

• But the hand moves 7× further than the biceps shortens

Example: Calf Raise (Standing on One Leg)

• Achilles tendon inserts 5 cm from ball of foot (fulcrum)

• Body weight acts 12 cm from ball of foot (at ankle)

$MA = \frac{EA}{RA} = \frac{5}{12} = 0.42$

Wait — this seems like MA < 1, but calf raise is second class (should be > 1)?

Correction: In second-class levers, EA is measured from fulcrum to effort, and RA from fulcrum to resistance. Let's reconsider:

• Fulcrum: Ball of foot

• Effort: Achilles tendon at heel (say 15 cm from ball of foot)

• Resistance: Body weight at ankle (say 10 cm from ball of foot)

$MA = \frac{EA}{RA} = \frac{15}{10} = 1.5$

Now MA > 1, confirming force advantage.

Trade-offs in Lever Systems

Why the Human Body Favors MA < 1

The human body predominantly uses third-class levers (MA < 1) because:

1. Speed is essential: Fast movements for locomotion, sports, survival

2. Range of motion is needed: Large movements from small muscle contractions

3. Muscle architecture: Muscles can generate high force but have limited shortening distance

4. Pennation angles: Many muscles are designed for force, not distance

5. Evolutionary advantage: Quick reactions more valuable than raw strength in many situations

Calculating Required Muscle Force

At equilibrium (lever balanced): $E \times EA = R \times RA$

Solving for effort: $E = \frac{R \times RA}{EA}$

Example: Holding 10 kg in Hand

• Weight = 10 kg × 9.8 = 98 N

• Biceps insertion = 5 cm from elbow

• Weight distance = 35 cm from elbow

$E = \frac{98 \times 35}{5} = \frac{3430}{5} = 686 \text{ N}$

The biceps must generate 686 N (70 kg equivalent) to hold just 10 kg in the hand!

Torque (Moment of Force)

Definition

Torque (also called moment of force or simply moment) is the rotational effect of a force. It is the measure of the tendency of a force to cause rotation about an axis or pivot point.

Formula

$\tau = F \times d$

Or more precisely: $\tau = F \times d \times \sin\theta$

Where:

• $\tau$ (tau) = torque (N·m)

• $F$ = force applied (N)

• $d$ = moment arm / lever arm (m)

• $\theta$ = angle between force vector and lever arm

When force is perpendicular to the lever arm, $\sin\theta = 1$, so $\tau = F \times d$.

Units

• SI unit: Newton-metres (N·m)

• Not the same as Joules, despite having the same dimensions

Moment Arm (Lever Arm)

Definition

The moment arm (also called lever arm or perpendicular distance) is the shortest distance from the axis of rotation to the line of action of the force. It is always measured perpendicular to the force vector.

Key Concept

The moment arm determines how effectively a force creates rotation:

• Larger moment arm = greater torque for same force

• Smaller moment arm = less torque for same force

• Moment arm = 0 (force through axis) = no torque

Force acting at distance d from pivot:

        Force (F)
           ↓
           │
    ←──d───│
           │
    ───────●───────
         Pivot
         
Torque = F × d

Moment Arm When Force is Not Perpendicular

        Force (F) at angle θ
             ↘
              \
    ←────d────●
              │
    ──────────●───────
            Pivot
            
Moment arm = d × sin(θ)
Torque = F × d × sin(θ)

Direction of Torque

By convention:

• Counter-clockwise (CCW): Positive torque

• Clockwise (CW): Negative torque

Or:

• Torque tending to rotate toward you (right-hand rule): Positive

• Torque tending to rotate away from you: Negative

Net Torque and Equilibrium

For rotational equilibrium: $\Sigma\tau = 0$ $\text{Sum of clockwise torques} = \text{Sum of counter-clockwise torques}$

This principle is used in:

• Lever analysis

• Balance and stability calculations

• Determining muscle force requirements

Torque in Human Movement

Muscle Torque

When a muscle contracts, it creates torque about a joint: $\tau_{muscle} = F_{muscle} \times d_{insertion}$

Where $d_{insertion}$ is the perpendicular distance from the joint axis to the muscle's line of pull.

Resistance Torque

External loads create resistance torque: $\tau_{resistance} = F_{resistance} \times d_{resistance}$

Equilibrium Condition

To hold a position statically: $\tau_{muscle} = \tau_{resistance}$ $F_{muscle} \times d_{muscle} = F_{resistance} \times d_{resistance}$

Worked Examples

Example 1: Biceps Curl (Static Hold)

An athlete holds a 15 kg dumbbell with the forearm horizontal. The dumbbell is 30 cm from the elbow, and the biceps inserts 4 cm from the elbow.

Find the biceps force required.

Given:

• Weight = 15 × 9.8 = 147 N

• Weight moment arm = 0.30 m

• Biceps moment arm = 0.04 m

Resistance torque: \tau_R = 147 \times 0.30 = 44.1 \text{ N·m}

For equilibrium: $\tau_{biceps} = \tau_R$ $F_{biceps} \times 0.04 = 44.1$ $F_{biceps} = \frac{44.1}{0.04} = 1102.5 \text{ N}$

The biceps must generate over 1100 N (equivalent to lifting ~112 kg) to hold just 15 kg!

Example 2: Effect of Joint Angle

The same biceps curl, but now the elbow is at 45° (forearm not horizontal).

The weight still acts vertically, but its moment arm changes: $d_{weight} = 0.30 \times \cos(45°) = 0.30 \times 0.707 = 0.212 \text{ m}$

(This assumes the forearm is at 45° to horizontal, so the horizontal distance to the weight is reduced.)

New resistance torque: \tau_R = 147 \times 0.212 = 31.2 \text{ N·m}

New biceps force: $F_{biceps} = \frac{31.2}{0.04} = 780 \text{ N}$

The biceps force required is reduced because the weight's moment arm is smaller!

This is why exercises are harder at certain joint angles (typically when the resistance moment arm is maximized).

Example 3: Seesaw Balance

Two children are on a seesaw. Child A (30 kg) sits 2.5 m from the fulcrum. Where must Child B (40 kg) sit for balance?

For equilibrium: $\tau_A = \tau_B$ $m_A \times g \times d_A = m_B \times g \times d_B$ $30 \times 2.5 = 40 \times d_B$ $d_B = \frac{75}{40} = 1.875 \text{ m}$

Child B must sit 1.875 m from the fulcrum.

Torque in Sport Performance

Throwing

• Greater moment arm (longer lever) = greater torque on thrown object

• Sequential segment rotation creates cumulative torques

• Proximal to distal sequence maximizes end velocity

Striking (Golf, Baseball, Tennis)

• Club/bat/racket length determines moment arm

• Longer implement = greater torque on ball (but harder to control)

• Sweet spot location affects torque transfer

Joint Loading

• Heavy loads at distance create large joint torques

• This is why deadlift form matters (keep load close to axis)

• Improper form increases injury risk due to excessive torque

Torque and Angular Acceleration

Newton's Second Law for Rotation

$\tau = I\alpha$

Where:

• $\tau$ = net torque (N·m)

• $I$ = moment of inertia (kg·m²)

• $\alpha$ = angular acceleration (rad/s²)

This is the rotational equivalent of F = ma.

Implications

• Greater torque = greater angular acceleration (for same I)

• Greater moment of inertia = less angular acceleration (for same torque)

• Athletes generate torque to create rotation

Example: Golf Swing

A golfer generates 100 N·m of torque on a club with moment of inertia 0.4 kg·m².

$\alpha = \frac{\tau}{I} = \frac{100}{0.4} = 250 \text{ rad/s}^2$

This high angular acceleration is what produces high club head speeds.

Practical Applications Summary

Stability Applications

Lever Applications

Torque Applications

Key Equations Summary

Exam Tips

1. Know lever classes: Remember E-F-R arrangement and which provides force vs speed advantage

2. Third class is most common in body: Prioritizes speed/ROM over force

3. MA calculations: MA = EA/RA; know what MA > 1 and MA < 1 mean

4. Torque = Force × Moment arm: The perpendicular distance is crucial

5. Stability factors: Lower COM, larger BOS, centered COM = more stable

6. COM can be outside body: Important for high jump, diving positions

7. Moment arm changes with joint angle: This affects muscle force requirements

8. Calculate muscle forces: Use torque equilibrium (muscle torque = resistance torque)

9. Apply to sport examples: Be ready to explain how levers and torque affect performance

10. Draw diagrams: Visualize the lever system, forces, and moment arms