Moment of Inertia

Overview

This topic explores the rotational equivalents of mass and momentum. While mass determines how difficult it is to accelerate an object linearly, moment of inertia determines how difficult it is to accelerate an object rotationally. Understanding moment of inertia and angular momentum conservation is essential for analyzing spinning, somersaulting, twisting, and any sport skill involving rotation. Athletes manipulate these principles constantly — often instinctively — to control their rotation speed during performance.

Moment of Inertia (I)

Definition

Moment of inertia is the resistance of an object to changes in its rotational motion. It is the rotational equivalent of mass. Just as mass resists linear acceleration, moment of inertia resists angular acceleration.

Conceptual Understanding

• A high moment of inertia means the object is difficult to start rotating, stop rotating, or change its rotation speed

• A low moment of inertia means the object is easy to start rotating, stop rotating, or change its rotation speed

• Moment of inertia depends on both mass AND the distribution of that mass relative to the axis of rotation

Mathematical Definition

For a point mass: $I = mr^2$

For a system of particles or extended body: $I = \Sigma mr^2 = m_1r_1^2 + m_2r_2^2 + m_3r_3^2 + ...$

Where:

• I = moment of inertia (kg·m²)

• m = mass (kg)

• r = perpendicular distance from the axis of rotation (m)

Units

• SI unit: Kilogram metres squared (kg·m²)

• No equivalent named unit exists

Key Factors Affecting Moment of Inertia

1. Total Mass

• Greater mass = greater moment of inertia

• Doubling the mass doubles the moment of inertia (linear relationship)

• This is why heavier athletes may rotate more slowly than lighter athletes (assuming similar body positions)

2. Distribution of Mass Relative to the Axis

• Mass further from the axis contributes MORE to moment of inertia

• Mass closer to the axis contributes LESS to moment of inertia

• The relationship is squared (r²), making distance very important

Critical insight: Moving mass from 1 m to 2 m from the axis increases that mass's contribution to moment of inertia by a factor of 4 (2² = 4), not 2!

3. Axis of Rotation

• The same object has different moments of inertia about different axes

• The axis must be specified when discussing moment of inertia

• Moment of inertia is always calculated relative to a specific axis

Visualising the r² Effect

Axis of Rotation
      |
      |    ●──────────────● m at distance 2r
      |    |              
      |    |  ●────● m at distance r
      |    |  |    
      |    |  |    
      
If mass m is at distance r: I = mr²
If mass m is at distance 2r: I = m(2r)² = 4mr²

Moving mass twice as far = 4× contribution to I

Moment of Inertia for Common Shapes

Human Body Moment of Inertia

The human body is not a simple geometric shape, so moment of inertia is determined through:

1. Mathematical modelling: Dividing body into segments, calculating each segment's contribution

2. Experimental measurement: Using specialized equipment

3. Estimation: Using published data and body composition

Typical values for adult male about different axes:

Key observation: Moment of inertia can change by a factor of 3-5 just by changing body position!

Calculating Moment of Inertia: Worked Examples

Example 1: Simple System

Two masses of 2 kg each are connected by a massless rod 1 m long. Calculate the moment of inertia about an axis through the centre of the rod.

      Axis
        |
2 kg ●──|──● 2 kg
   0.5m | 0.5m

Solution: Each mass is 0.5 m from the axis.

I = m_1r_1^2 + m_2r_2^2 = 2(0.5)^2 + 2(0.5)^2 = 0.5 + 0.5 = 1.0 \text{ kg·m}^2

Example 2: Effect of Mass Distribution

Using the same system, calculate I if the axis is at one end.

    Axis
      |
2 kg ●────────● 2 kg
      |   1m
      0         1m from axis

Solution: I = m_1r_1^2 + m_2r_2^2 = 2(0)^2 + 2(1)^2 = 0 + 2 = 2.0 \text{ kg·m}^2

Comparison: Same masses, same rod, but different axis location → different moment of inertia (2.0 vs 1.0 kg·m²)

Example 3: Gymnast Changing Position

A gymnast (mass 60 kg) changes from layout to tuck position. Estimate the change in moment of inertia about the transverse axis.

Given estimates:

• Layout: I ≈ 14 kg·m²

• Tuck: I ≈ 4 kg·m²

Change: \Delta I = 4 - 14 = -10 \text{ kg·m}^2

The moment of inertia decreases by approximately 71% (from 14 to 4 kg·m²)

Example 4: Figure Skater's Arms

A figure skater's arms (total mass 8 kg, treated as point masses at the hands) are initially extended 0.8 m from the body's longitudinal axis. Calculate the contribution to moment of inertia, then recalculate when arms are pulled in to 0.1 m.

Arms extended: I_{arms} = mr^2 = 8 \times (0.8)^2 = 8 \times 0.64 = 5.12 \text{ kg·m}^2

Arms tucked: I_{arms} = mr^2 = 8 \times (0.1)^2 = 8 \times 0.01 = 0.08 \text{ kg·m}^2

Change factor: $\frac{5.12}{0.08} = 64$

The arms' contribution to moment of inertia is 64 times greater when extended compared to when tucked!

Angular Momentum (L)

Definition

Angular momentum is the quantity of rotational motion possessed by a rotating body. It is the rotational equivalent of linear momentum.

Mathematical Expression

$L = I\omega$

Where:

• L = angular momentum (kg·m²/s)

• I = moment of inertia (kg·m²)

• ω = angular velocity (rad/s)

Units

• SI unit: Kilogram metres squared per second (kg·m²/s)

• Sometimes expressed as Newton metre seconds (N·m·s) or Joule seconds (J·s)

Characteristics

• Vector quantity: Has magnitude and direction

• Direction: Same as angular velocity (along the axis of rotation, following right-hand rule)

• Depends on both I and ω: Can have same angular momentum with different combinations of I and ω

Comparison: Linear vs Angular Momentum

Calculating Angular Momentum

Example 1: A diver in tuck position (I = 4 kg·m²) rotates at 12 rad/s. Calculate angular momentum.

L = I\omega = 4 \times 12 = 48 \text{ kg·m}^2\text{/s}

Example 2: A figure skater (I = 2.5 kg·m²) spins at 25 rad/s. Calculate angular momentum.

L = I\omega = 2.5 \times 25 = 62.5 \text{ kg·m}^2\text{/s}

Conservation of Angular Momentum

The Law of Conservation of Angular Momentum

In the absence of external torques, the total angular momentum of a system remains constant.

$L_{initial} = L_{final}$

$I_1\omega_1 = I_2\omega_2$

Conditions for Conservation

Angular momentum is conserved when:

1. No external torques act on the system (the system is isolated rotationally)

2. Internal forces/torques can change distribution but not total angular momentum

When Is There No External Torque?

• Airborne athletes: No external torque once feet leave the ground (gravity acts through centre of mass, so no torque)

• Frictionless spinning: Ideal spinning on ice or very low friction surface

• Free rotation: Any rotation where no external force creates a turning effect

Mathematical Derivation

From Newton's Second Law for rotation: $\tau = \frac{dL}{dt}$

If τ = 0 (no external torque): $\frac{dL}{dt} = 0$

Therefore L = constant

The Fundamental Relationship

$I_1\omega_1 = I_2\omega_2 = \text{constant}$

This can be rearranged to: $\frac{\omega_2}{\omega_1} = \frac{I_1}{I_2}$

Key insight: If moment of inertia decreases, angular velocity must increase proportionally, and vice versa.

How Athletes Change Rotation Speed

The Mechanism

Since angular momentum is conserved during flight (no external torque), athletes can only change their rotation speed by changing their moment of inertia.

$I\omega = \text{constant}$

• To rotate FASTER: Decrease moment of inertia (bring mass closer to axis)

• To rotate SLOWER: Increase moment of inertia (move mass away from axis)

Inverse Relationship

If I decreases by factor of 2 → ω increases by factor of 2 If I decreases by factor of 4 → ω increases by factor of 4

Specific Techniques

Somersaulting (Rotation About Transverse Axis)

Practical application:

• Diver: Takes off in layout or pike, tucks to complete rotations quickly, opens to layout for entry

• Gymnast: Uses tuck for multiple somersaults, layout for single somersaults with aesthetic appeal

• Trampolinist: Manipulates position throughout skill to control rotation

Twisting (Rotation About Longitudinal Axis)

Practical application:

• Figure skater: Arms out to start spin slowly, pull arms in to spin rapidly

• Diver: Initiates twist with asymmetric arm position, pulls arms in to twist faster

• Gymnast: Arms wide during takeoff, pulled in for rapid twisting

Combined Rotations

Athletes performing simultaneous somersaults and twists must manage angular momentum about multiple axes:

• Somersault momentum: Generated at takeoff, cannot be changed

• Twist momentum: Can be generated in flight through body asymmetries

• Trade-off: Increasing rotation about one axis may decrease rotation about another

Worked Examples: Conservation of Angular Momentum

Example 1: Figure Skater Spin

A figure skater begins a spin with arms extended (I = 3.5 kg·m²) at 2 rev/s. She pulls her arms in (I = 1.0 kg·m²). Calculate her new spin rate.

Solution: Using conservation of angular momentum: $I_1\omega_1 = I_2\omega_2$

$3.5 \times 2 = 1.0 \times \omega_2$

$\omega_2 = \frac{7}{1} = 7 \text{ rev/s}$

The skater's spin rate increases from 2 rev/s to 7 rev/s — a 3.5× increase!

Example 2: Diver Somersaulting

A diver leaves the board in layout position (I = 15 kg·m²) with an angular velocity of 4 rad/s. She tucks (I = 4 kg·m²) during flight. Calculate: a) Her angular velocity in tuck b) The factor by which her rotation speed increased

Solution: a) Using conservation: $I_1\omega_1 = I_2\omega_2$ $15 \times 4 = 4 \times \omega_2$ $\omega_2 = \frac{60}{4} = 15 \text{ rad/s}$

b) Factor of increase: $\frac{\omega_2}{\omega_1} = \frac{15}{4} = 3.75$

Her rotation speed increases by a factor of 3.75 (or 275% increase)

Example 3: Gymnast Preparing for Landing

A gymnast performing a tucked double somersault (I = 4 kg·m², ω = 14 rad/s) opens to pike (I = 7 kg·m²) before landing. Calculate the new angular velocity.

Solution: $I_1\omega_1 = I_2\omega_2$ $4 \times 14 = 7 \times \omega_2$ $\omega_2 = \frac{56}{7} = 8 \text{ rad/s}$

The gymnast slows from 14 rad/s to 8 rad/s, allowing for a controlled landing.

Example 4: Angular Momentum During Dive

A diver generates 55 kg·m²/s of angular momentum at takeoff. If she performs the dive in tuck (I = 3.8 kg·m²), pike (I = 6.5 kg·m²), and layout (I = 13 kg·m²) positions, calculate the angular velocity in each.

Solution: Since L = Iω and L = 55 kg·m²/s (constant):

Tuck: $\omega = \frac{L}{I} = \frac{55}{3.8} = 14.5 \text{ rad/s}$

Pike: $\omega = \frac{L}{I} = \frac{55}{6.5} = 8.5 \text{ rad/s}$

Layout: $\omega = \frac{L}{I} = \frac{55}{13} = 4.2 \text{ rad/s}$

Applications Across Sports

Diving

Typical Dive Phases:

1. Takeoff: Generate angular momentum; position determines initial I and ω

2. Flight (tuck/pike): Decrease I to increase ω for faster rotation

3. Pre-entry: Increase I (open out) to decrease ω for controlled entry

4. Entry: Streamlined position for minimal splash

Dive Complexity:

• More rotations require: Greater initial angular momentum OR more extreme tuck

• Elite divers: Can complete 4.5 somersaults (1620°) in platform diving

• Limiting factors: Takeoff angular momentum, ability to tuck tightly, time in air

Gymnastics

Floor Exercise:

• Multiple somersaults with twists

• Angular momentum generated during takeoff

• Body position changes control rotation speed

• Must land with appropriate angular velocity (too fast = overrotation)

Uneven Bars/High Bar:

• Giants (full rotations around bar) demonstrate angular momentum

• Release moves require precise angular momentum

• Catching requires matching angular velocity

Balance Beam:

• Limited space for generating angular momentum

• Precise control of rotation essential

• Dismounts require angular momentum management

Figure Skating

Spins:

• Upright spin: Start slow (arms out), accelerate (arms in)

• Sit spin: Lower position changes moment of inertia

• Camel spin: Horizontal leg position affects I

• Combination spins: Change position multiple times, varying speed

Jumps:

• Single/double/triple/quadruple rotations

• Angular momentum generated at takeoff

• Arms pulled in immediately for fast rotation

• Must complete rotation and open before landing

Elite Examples:

• Quadruple jumps require ~3.5-4 rev in ~0.7 s airtime

• That's approximately 5-6 rev/s or 30-35 rad/s!

Trampolining

Multiple Somersaults:

• High bounce provides long airtime

• Tuck position allows rapid rotation

• Opening controls rotation for landing

• Elite athletes perform triple somersaults with multiple twists

Springboard/Platform Diving

Height Considerations:

• 3m springboard: ~0.9-1.0 s airtime for forward dives

• 10m platform: ~1.4 s airtime

• More airtime = more time to rotate = more complex dives possible

• Angular momentum must be greater for more rotations but still controllable

Ski Aerials

Requirements:

• Generate angular momentum during ramp departure

• Control rotation during ~3 s airtime

• Land with minimal rotation on steep slope

• Triple somersaults with multiple twists performed by elite athletes

Torque and Angular Momentum

Relationship

Torque is the rate of change of angular momentum:

$\tau = \frac{dL}{dt}$

For constant torque: $\tau = \frac{\Delta L}{\Delta t}$

Generating Angular Momentum

To generate angular momentum, an external torque must be applied:

• Ground contact: Force applied away from centre of mass creates torque

• Duration of contact: Longer contact time = greater change in angular momentum

• Force magnitude: Greater force = greater torque = faster change in momentum

Angular Impulse

Angular impulse equals change in angular momentum:

$\tau \times \Delta t = \Delta L$

Where:

• τ × Δt = angular impulse (N·m·s)

• ΔL = change in angular momentum (kg·m²/s)

Sport Applications

Takeoff for Rotation:

Radius of Gyration

Definition

The radius of gyration (k) is the distance from the axis of rotation at which all the mass of a body could be concentrated without changing its moment of inertia.

Mathematical Expression

$I = mk^2$

Therefore: $k = \sqrt{\frac{I}{m}}$

Significance

• Provides a single distance value representing mass distribution

• Smaller k = mass concentrated closer to axis = lower I

• Larger k = mass distributed further from axis = higher I

Sport Applications

• Athletes with longer limbs have larger radii of gyration

• Body position changes alter the radius of gyration

• Useful for comparing athletes of different sizes

Factors Affecting Performance

Body Type Considerations

Optimizing Rotation

For Faster Rotation:

1. Generate more angular momentum at takeoff

2. Assume a tighter tuck/body position (lower I)

3. Pull mass as close to axis as possible (arms in, chin tucked, knees to chest)

For Controlled Landing:

1. Open body position to increase I

2. This decreases ω to a manageable level

3. Time the opening appropriately

Trade-offs

Common Misconceptions

Misconception 1: "Athletes spin faster by moving their arms"

Reality: Moving arms doesn't generate new angular momentum in flight; it redistributes existing momentum by changing I.

Misconception 2: "Angular momentum can be created during flight"

Reality: In the absence of external torque (during flight), total angular momentum is constant. It can only be redistributed, not created.

Misconception 3: "Heavier athletes can't rotate as fast"

Reality: While higher mass increases I, heavier athletes may generate more angular momentum at takeoff. The key is the ratio of angular momentum to moment of inertia.

Misconception 4: "Moment of inertia is the same as mass"

Reality: Moment of inertia depends on BOTH mass AND mass distribution relative to the axis. The same mass can have vastly different moments of inertia.

Summary of Key Equations

Exam Tips

1. Remember I = Σmr²: The r² term makes distance from axis critically important

2. Conservation only applies when there's no external torque: Typically during flight

3. Inverse relationship: When I decreases, ω increases proportionally

4. Be able to calculate: Both simple moment of inertia problems and conservation problems

5. Link to sport examples: Be ready to explain how athletes manipulate I to control ω

6. Understand the mechanism: Athletes cannot create or destroy angular momentum in flight — only redistribute it

7. Know body position effects: Tuck < Pike < Layout for moment of inertia about transverse axis

8. Apply to multiple sports: Diving, gymnastics, skating, trampolining, ski aerials