Physics of Torque, Angular Kinetics, and Body Stability in Biomechanics

What is torque?

The turning effect produced by a force about an axis of rotation; calculated as T = F × r, where F is force and r is the moment arm.

What are the types of lines of action for forces and their effects?

Centric force: Passes through the center of mass; causes linear motion only. Eccentric force: Does not pass through center of mass; causes both rotation and translation. Force couple: Equal and opposite forces acting at different points; causes pure rotation.

What is a moment arm?

The perpendicular distance between the line of action of the force and the axis of rotation.

What happens to torque as the moment arm increases?

Torque increases proportionally with moment arm length (T ∝ r).

What three components describe a torque?

Axis of rotation; Magnitude (strength of the turning effect); Direction (clockwise or counterclockwise, determined via right-hand rule).

What is the right-hand rule and how does it determine torque direction?

Curl the fingers of your right hand in the direction of rotation; your thumb points toward the positive torque direction (usually out of the page for counterclockwise, into the page for clockwise).

What is an example of an internal torque?

The torque created by muscle tension acting on bones around a joint (e.g., biceps generating elbow flexion torque).

Do moment arms of a muscle change during motion or contraction?

Yes, they change dynamically as the joint angle changes—affecting mechanical advantage and torque output.

What are the conditions for static equilibrium?

ΣF = 0 (sum of forces = 0); ΣT = 0 (sum of torques = 0); This means there is no linear or angular acceleration.

What is the equation for torque equilibrium?

ΣT = Tclockwise + Tcounterclockwise = 0.

Define center of gravity (CG).

The point at which the entire mass of a body can be considered to be concentrated; gravitational force acts downward through this point.

How does adding an object to the body affect CG?

CG shifts toward the added mass (e.g., holding an object in front moves CG forward).

What happens to the center of gravity when a body becomes a projectile?

The CG follows a fixed parabolic trajectory, regardless of body motion during flight.

What is stability?

The ability of a body to return to equilibrium after being displaced.

What factors increase stability?

Lowering the center of gravity; Increasing base of support; Aligning CG within the base of support; Increasing body mass.

What does angular kinematics describe?

The motion of bodies that rotate or spin about an axis without considering the forces causing the motion.

What is an angle?

The orientation between two intersecting lines, planes, or segments, measured in degrees or radians.

What is the difference between absolute and relative angular position?

Absolute angle: Orientation of a segment relative to a fixed line (e.g., horizontal). Relative angle: Angle between two adjacent segments (e.g., thigh and shank).

What is angular displacement?

The change in angular position of a rotating line or body, measured in degrees (°) or radians (rad).

What is the sign convention for angular direction?

Counterclockwise: Positive (+); Clockwise: Negative (−).

When an object rotates, do all points travel the same linear distance?

No, points farther from the axis of rotation travel a greater linear distance though the same angular distance.

What is the formula linking angular displacement and linear distance?

ℓ = Δθ × r, where ℓ is arc length, Δθ is angular displacement, and r is radius.

Define angular velocity.

The rate of change of angular displacement over time; ω = Δθ / Δt (units: rad/s or °/s).

How does linear velocity relate to angular velocity?

vT = ω × r, where vT is tangential velocity; points farther from the axis move faster linearly.

Define angular acceleration.

The rate of change of angular velocity over time: α = Δω / Δt.

How do you determine if something is speeding up or slowing down in rotation?

Compare the sign of angular velocity (ω) and angular acceleration (α): Same sign → speeding up; Opposite signs → slowing down.

What is tangential acceleration?

Linear acceleration tangent to the circular path; caused by changes in angular velocity.

What is centripetal acceleration?

Acceleration directed toward the axis of rotation, maintaining circular motion: ar = v² / r = ω² × r.

If ω is constant, how does centripetal acceleration relate to radius?

ar ∝ r (directly proportional).

If tangential velocity (vT) is constant, how does centripetal acceleration relate to radius?

ar ∝ 1/r (inversely proportional).

What are the three anatomical planes of motion?

Sagittal plane - divides body left/right (e.g., flexion, extension); Frontal plane - divides body front/back (e.g., abduction, adduction); Transverse plane - divides body top/bottom (e.g., rotation).

What are the three anatomical axes of rotation?

Mediolateral (frontal-horizontal) - for sagittal plane motions; Anteroposterior (sagittal-horizontal) - for frontal plane motions; Longitudinal (vertical) - for transverse plane motions.

Give common motion examples for each plane.

Sagittal: Bicep curl, forward lunge; Frontal: Jumping jack, side bend; Transverse: Spinal rotation, baseball swing.

What is angular inertia?

Resistance to change in angular motion; depends on both mass and how it's distributed about the axis.

How does angular inertia differ from linear inertia?

Linear inertia depends only on mass, while angular inertia depends on mass distribution relative to the axis.

What is moment of inertia (I)?

A measure of an object's angular resistance to rotation: I = m × k², where k is the radius of gyration.

What happens to moment of inertia if you double mass or radius of gyration?

Doubling mass → I doubles; Doubling radius of gyration → I increases by four times (since radius is squared).

How does moment of inertia change with radius change?

Moving mass farther from the axis (increasing radius) increases moment of inertia, making rotation harder.

Provide real-world examples of inertia change.

Choking up on a bat: Decreases radius → decreases I → easier to swing. Figure skater pulling arms in: Decreases I → spins faster.

Define angular momentum.

Quantifies angular motion and resistance to change: H = I × ω.

What is Newton's First Law in angular terms?

A rotating body continues in its state of angular motion unless acted upon by an external torque.

What happens when angular momentum (H) is constant?

↑ I → ↓ ω (slower rotation); ↓ I → ↑ ω (faster rotation).

How does conservation of angular momentum apply to movement?

Example: A skater spins faster when pulling arms in, slower when extending them.

What is Newton's Second Law in angular form?

If a net external torque acts on a body, it will angularly accelerate in the torque's direction: ΣT = I × α.

What does the magnitude of angular acceleration depend on?

Directly on torque magnitude, inversely on moment of inertia.

What does the change in angular momentum depend on?

It is proportional to the net external torque and occurs in the torque's direction.

What happens when torque and angular velocity have the same sign?

Angular momentum increases.

What happens when torque and angular velocity have opposite signs?

Angular momentum decreases.