Torque and Newton's Laws 8B

Announcements and Updates

• Lecture quiz results were excellent, with no need for in-class review of specific questions. Individual results are available for review.

• Today's focus is set 8B, covering torque, Newton's laws, and the biceps curl.

• Exam 3 will follow the format of Exams 1 and 2: multiple-choice and true/false questions based on lecture and practice set material.

Exam 3 Information

• The theta, omega, and alpha profiles for both the forearm segment angle and the included elbow joint angle during a biceps curl will be provided.

• PVNA plots of the barbell will not be provided directly, although linear kinetics and energy concepts remain relevant.

• The instructor plans to cover sets 9A and 9B before the exam, which focus on single-joint systems to understand individual muscle torque production.

• There will be no lecture quiz due next week to allow focus on exam preparation.

Torpedo Bat Discussion

• A new "torpedo bat" design was introduced, featuring a thicker cross-section in the primary contact area that tapers towards the end, contrasting traditional bats with uniform width after widening.

• The modified bat design is legal, with the shape change intended to increase mass in the hitting zone and enlarge the contact surface area.

• How might this change alter the moment of inertia of the bat about the grip end?

  • The moment of inertia decreases because mass is relocated closer to the axis of rotation (grip end), reducing the r^2 terms in the moment of inertia calculation.

  • This should make the bat easier to swing and increase angular acceleration and bat speed.

  • The primary performance enhancement is believed to be a larger hitting area and favorable center of percussion.

Angular Kinetics and Newton's Laws

• The lecture returns to angular kinetics, focusing on how torque integrates Newton's laws into angular mechanics (kinematics and kinetics).

• All three of Newton's laws are relevant in this context.

• Mass moment of inertia is crucial, requiring consideration of mass distribution relative to the axis of rotation.

Mass Distribution in Limbs

• The mass of human long segments (arms, legs) is typically distributed closer to the proximal end.

• This distribution makes rotation about the proximal end easier due to lower moment of inertia.

• In throwing, striking, and kicking, this mass distribution facilitates high angular velocities and accelerations.

• Reducing the moment of inertia I allows more torque \tau to contribute to angular acceleration \alpha ($\$\tau = I \alpha\$).

Static vs. Dynamic Torque

• Static component: Torque due to limb weight and moment arm.

• Exists even when moving (dynamically).

• Reducing R_wW (moment arm times weight) reduces resistance against muscle action.

• More net torque can be used for angular acceleration.

Newton's First Law (Angular Inertia)

• A rotating body will continue in a state of uniform angular motion unless acted on by an external torque.

• Mathematically represented as: \sum \tau = 0.

• Uniform angular motion implies constant angular momentum.

• Angular momentum (L) is calculated as: L = I \omega, where I is the moment of inertia and \omega is the angular velocity.

• If I changes, \omega must adjust to keep L constant when \sum \tau = 0.

• Angular acceleration can occur without applied torque by altering the moment of inertia.

Conservation of Angular Momentum Examples

• Figure skater spinning: By drawing arms and legs inward, the skater reduces their moment of inertia (I), causing an increase in angular velocity (\omega) to conserve angular momentum (L).

• Divers: By tucking (reducing I), divers increase their angular velocity; extending (increasing I) slows rotation. Gravity pulls through their center of mass, thus having no external torque.

Newton's Second Law (Angular Kinetics)

• Analogous to F = MA, the angular form is: \sum \tau = I \alpha, where \sum \tau is the sum of external torques, I is the moment of inertia, and \alpha is the angular acceleration.

• For this equation to hold, I must be constant.

• Torques can always be summed about the object's center of mass.

• Torques can be summed about an axis other than the center of mass only if the axis is linearly fixed (pure rotation).

• Angular acceleration must be in radians per second squared (rad/s^2) for calculations.

• Constant moment of inertia implies a rigid body (no change in size or shape).

Implications of Angular Acceleration

• A negative value of \alpha indicates acceleration opposite to the drawn direction in the diagram.

• For a rigid object, angular acceleration is uniform throughout the plane, regardless of the summation point.

Free Body Diagrams and Equations of Motion

• Two equations of motion can be applied: linear (\sum F = MA) and angular (\sum \tau = I \alpha).

• Mass must be constant for \sum F = MA to be valid, \and I must be constant for \sum \tau = I \alpha to be valid.

• Even if I is not constant, the linear equation of motion can still be used.

• The right-hand side of the angular equation (\tau = I \alpha) is referred to as the inertial torque.

Biceps Curl Example: Summing Torques About the Center of Mass

• When summing about the center of mass, forces like the bone-on-bone force at the elbow joint, originally dismissed, now factor in due to a moment arm relative to the center of mass.

• The equation becomes complex but remains mathematically valid.

Biceps Curl Example: Pure Rotation at the Elbow

• In pure rotation, torques can be summed about the elbow joint: \sum \tau{elbow} = I{elbow} \alpha.

• Muscle torque is positive (counterclockwise), weight torque (WRW) is negative, and the hand force torque is negative.

Simplified Biceps Curl Analysis

• The barbell weight can be merged into the forearm and hand system for simplification.

• The total system weight is the sum of forearm/hand and barbell weights.

• The total moment of inertia is the sum of forearm/hand and barbell moments of inertia.

Benefits of Simplification

• The simplified equation highlights that muscle torque offsets the torque due to the weight of the system, with any remaining torque contributing to angular acceleration.

• The force at the hand is now internal to the system and not considered a torque producer.

Limitations of the Simplified Model

• The model assumes only one muscle is responsible for torque generation, which isn't realistic.

• Multiple muscles (biceps, brachialis, brachioradialis, pronator teres) contribute to elbow flexion.

• Passive elements (ligaments, fascia) and antagonists can also produce torques.

Generalized Free Body Diagram

• A technique to simplify the free body diagram while accounting for multiple forces.

• Forces crossing the joint are grouped into a single joint reaction force.

• The torque produced by these forces about the elbow joint is accounted for by a net muscular moment.

Net Muscular Moment

• The sum of all torque-producing elements, including muscles, bone-on-bone force, ligaments, and fascia, at the elbow joint.

• Mathematically represented as:

  • \sum (Elbow \, Flexors) - \sum (Elbow \, Extensors) + \sum (Passive \, Components)

• Can represent flexor or extensor torques without focusing on individual muscles.

Angular Equation of Motion with Generalized Free Body Diagram

• The equation becomes: NMM - WRW = I \alpha

• NMM: Net Muscular Moment (torque produced by muscles)

• WRW: Torque due to weight

• I \alpha: Inertial torque

• This equation includes any combination of flexor, extensor, and passive tissues creating torques at the joint.

• It still accounts for weight and inertial torque.

Linear Equation of Motion with Generalized Free Body Diagram

• Incorporates force arrows without addressing the torque arrow.

• Addresses the forces in horizontal and vertical directions.

Torque Profiles Under Different Conditions

• Net muscular moment at the elbow during a biceps curl under quasi-static conditions:

  • Linear and angular acclerations are zero.

  • Equation reduces to NMME = WRW.

  • Torque is greatest when weight is furthest horizontally from the elbow joint.

  • Uniphasic activation pattern, with flexor torque needed throughout the entire up and down phases.

• Net muscular moment at the elbow during a biceps curl under dynamic conditions:

  • I \alpha is not zero, but is smaller than WRW.

  • Adding I \alpha$$ and WRW terms together.

  • Constant I, thus I can be constant only here.

  • Hill valley valley hill profile.