Angular Kinetics: Torques, Statics, and Dynamics

Fundamental Principles of Torque

• Definition of Torque: A torque is created by a force acting at a distance away from the axis of rotation.

• Gravitational Torques: Gravity produces torque when the line of gravity does not pass through the pivot point (e.g., the hip joint).     

• Trunk Flexion: Gravity acts on the trunk to produce a clockwise torque about the lumbar vertebrae.     

• Lateral Arm Raise: The weight of the arm and a dumbbell produces a clockwise torque about the shoulder joint.     

• Muscular Counteraction: In order to hold these positions, a muscular torque in the opposite (counterclockwise) direction must counteract the gravitational torques.

• Contact Forces: Torques are generated about the Center of Mass (CoM) during activities like diving and gymnastics

  • This involves utilizing vertical Ground Reaction Force (GRF) and specific body configurations that shift the CoM in front of or behind the point of force application.     

    • Example: In a back-flip, GRF is applied at a distance from the CoM, resulting in a clockwise rotation.

• Muscle Forces: Muscles also generate torques about joint centers to facilitate movement.

Analyzing Torques through Newton’s Three Laws

• Torque analysis is categorized into three frameworks based on the duration and nature of the application:     

• Instant in Time: Examines the immediate effect of torque.    

• Over a Period of Time: Examines the impulse and change in momentum.    

• Over a Distance: Examines mechanical angular work, power, and energy.

Torque at an Instant in Time: Statics

• General Formula: $\Sigma T = I\alpha$

• Static Condition: Occurs when angular acceleration is zero ($\alpha = 0$). Systems are either at rest or moving at a constant velocity.

• Equilibrium: State of balance that occurs when all rotations sum to zero: $\Sigma T_{system} = 0$

• Balancing Torques Example:     

• Force A: $670\,N$ at $2.3\,m$ from axis ($+\text{rotation}$).

  • Torque $T_A = 670\,N \times 2.3\,m = 1541\,N\text{-}m$.     

• Force B: $541\,N$ at $2.85\,m$ from axis ($-\text{rotation}$).

  • Torque $T_B = 541\,N \times 2.85\,m = 1541\,N\text{-}m$.   

• $\Sigma T = T_A - T_B = 1541 - 1541 = 0$ . The system remains in a balanced position.

• Multiple Internal Torques (Human Statics): For an individual holding a barbell at the elbow (axis of rotation):

  • Negative Torques (Clockwise): Forearm-hand system weight ($45\,N$ at $0.15\,m$) and Barbell weight ($420\,N$ at $0.4\,m$).

  • Positive Torque (Counterclockwise): Muscle force acting across the elbow flexors.     - Calculation of Total Torque required:       $T_{\text{arm-hand}} = 45\,N \times 0.15\,m = 6.75\,N\text{-}m$         $T_{\text{barbell}} = 420\,N \times 0.4\,m = 168\,N\text{-}m$        $T_{\text{muscle}} = 6.75\,N\text{-}m + 168\,N\text{-}m = 174.75\,N\text{-}m$

• Muscle Force Requirements: If the moment arm of the elbow flexors is $0.05\,m$, the required muscle force is: $F_{\text{muscle}} = \frac{174.75\,N\text{-}m}{0.05\,m} = 3495\,N$.     

  • The muscle force must be significantly larger than the weight of the objects because its moment arm is much smaller than those of the weights.

• Directional Dynamics:  

  • If the net torque is positive, rotation is counterclockwise (flexion).     

  • If the net torque is zero, the muscle action is isometric.

Influence of Joint Angle on Statics

• Moment arms change as the joint flexes or extends.

• When the arm is extended (angled below the horizontal), the moment arm is calculated using trigonometry:     

• $a = d \times \cos(\theta)$     

• At $0^{\circ}$ (parallel to the ground), the moment arm is at its maximum ($d$).     

• At $90^{\circ}$ (full extension), the moment arm is $0$ because the force line of action passes through the axis.

• Example Comparison (Barbell Curl):     

• Arm at $25^{\circ}$ below horizontal:     

  • Forearm moment arm: $0.15\,m \times \cos(25^{\circ}) = 0.15 \times 0.9063 = 0.14\,m$.     

  • Barbell moment arm: $0.4\,m \times \cos(25^{\circ}) = 0.4 \times 0.9063 = 0.36\,m$.     

• Total Muscle Force needed: $(45\,N \times 0.14\,m) + (420\,N \times 0.36\,m) = (6.3 + 151.2) \div 0.05 = 3150\,N$.

• Mechanical Disadvantage: Joints are typically third-class levers. This means muscles must exert massive forces due to short moment arms, but they allow for a magnified range of motion (ROM).

Stability and Balance

• Stability: The resistance to disruption, including both linear and angular acceleration.

• Balance: The ability to control equilibrium and maintain stability.

• Strategies:     

  • Maximizing Stability: Necessary for sports like wrestling or judo.     

  • Minimizing Stability: Necessary for sprinting or swimming to allow fast acceleration.

• Three Conditions of Equilibrium:    

  • Stable Equilibrium: Object returns to its original position after displacement (e.g., child on a swing, concave surfaces).     

  • Unstable Equilibrium: Object continues to move away from the original position (e.g., headstand, convex surfaces).    

  • Neutral Equilibrium: Object stops and assumes a new position without returning or moving further away (e.g., ball on a flat surface).

Factors Affecting Stability

• Mass: Higher mass requires higher force for acceleration ($F=ma$).

• Friction: Greater friction increases the force required to initiate motion (e.g., batting gloves increase grip stability).

• Base of Support (BoS): The area enclosed by the outer edges of the body in contact with the ground.     

  • Stability is lost when the line of action directed from the CoM moves outside the BoS.  

  • Widening the stance increases BoS in the mediolateral direction.

• Horizontal Position of CoM: Equilibrium is most stable when the line of gravity is in the center of the BoS.

• Height of CoM: A lower CoM increases stability. Higher CoM creates a greater disruptive torque during angular displacement.

• Biological/Clinical Factors:     

  • Mobility aids like walkers increase BoS.     

  • Fall risks in older adults include friction levels below $0.82$, decreased muscle tension, and impairments in strength, joint movement, hearing, vision, and cognition.

Torque at an Instant in Time: Dynamics

• Dynamic relationship: $\Sigma T = I\alpha$

• In a 2D system, angular acceleration ($\alpha_z$) occurs about the z-axis.     

  • If $\alpha_z = 0$, motion is linear.     

    • If $\alpha_x = 0$ and $\alpha_y = 0$, motion is rotational.     

      • Static case exists when all components are zero.

• Inverse Dynamics: Evaluates each segment starting from the most distal point and working proximally to calculate net joint moments.

Torque Over a Period of Time: Angular Impulse

• For rotation to occur, torques must be applied over time.

• Angular Impulse Formula: $T \times t = \Delta L = I\omega_{\text{final}} - I\omega_{\text{initial}}$

• Example: A gymnast hitting a vaulting horse converts horizontal linear velocity into vertical velocity and angular momentum via contact torques.

Torque Applied Over a Distance: Work and Energy

• Mechanical Angular Work ($W$): Product of torque and angular distance rotated.     - $W = T \times \Delta\theta$     - Units: Joules ($J$).     

  • Example: $40.5\,N\text{-}m \times 0.79\,rad = 32.0\,J$.

• Muscle Work Types:     

  • Positive Work: Associated with concentric muscle action (muscle shortening).     

  • Negative Work: Associated with eccentric muscle action (muscle lengthening).

• Angular Power ($P$): The rate of doing work.     

• $P = \frac{\Delta W}{\Delta t}$ or $P = T \times \omega$    

•  Units: Watts ($W$).

• Muscle Power Evaluation: $P_{\text{muscle}} = M_{\text{joint}} \times \omega_{\text{joint}}$     

  • Positive Power: Flexor moment with flexion OR extensor moment with extension (Concentric).     

  • Negative Power: Flexor moment with extension OR extensor moment with flexion (Eccentric).

• Rotational Kinetic Energy (RKE):     

• $\text{RKE} = \frac{1}{2}I\omega^2$     

• Total Energy: $\text{TE} = \text{KE} + \text{PE} + \text{RKE}$

• Work-Energy Relationship: $\Delta\text{Work} = \Delta\text{Energy}$     

• In baseball batting, energy is generated at contact; the bat also stores potential energy (PE) in the handle which transfers to kinetic energy at impact.

Questions & Discussion

• Biceps Force Calculation: To support a weight of $70\,N$ held at $30\,cm$ ($0.3\,m$) from the elbow, with the biceps attached at $3\,cm$ ($0.03\,m$):     

  • $T_{\text{weight}} = 70\,N \times 0.3\,m = 21\,N\text{-}m$     

  • $F_{\text{biceps}} = \frac{21\,N\text{-}m}{0.03\,m} = 700\,N$

• Frictionless Door Balance: Two individuals apply force to opposite sides of a door.    

  • Person A: $30\,N$ at $40^{\circ}$ angle, $45\,cm$ from hinge.     

  • Person B: Force at $90^{\circ}$ angle, $38\,cm$ from hinge.     

  • Solution involves setting $\Sigma T = 0$ to find the force applied by Person B.

• Stability Review:     

• Factors affecting stability include mass, friction, and base of support.     

• True/False: Neutral equilibrium is when an object returns to its original position. (False - that is stable equilibrium).     

• True/False: More mass means more stability. (True).

• Angular Work Calculation: If a $32\,N$ force is applied $0.2\,m$ from the axis and the object moves $12^{\circ}$.     

• Convert degrees to radians: $12 \times (\frac{\pi}{180}) = 0.2094\,rad$.     

• $T = 32 \times 0.2 = 6.4\,N\text{-}m$ .     - $W = 6.4 \times 0.2094 = 1.34\,J$.

• Knee Joint Power Calculation: Extensor torque of $60\,N\text{-}m$, interval stance phase flexion moves from $8.02^{\circ}$ to $14.9^{\circ}$ in $0.02\,s$.