Angular Kinematics and Torque

Angular Kinematics Discussion

  • Relating linear and angular stuff to each other

  • Pure rotation: points on an object rotate with the same angular displacement, but different linear displacements based on their distance from the axis of rotation.

Arc Length

  • The relationship between arc length (s), radius (r), and angular displacement \theta: s = r \theta Where:

    • s is the length of the arc on the circular path.

    • r is the radius of the circular path.

    • \theta is the angular displacement in radians.

  • This equation is a rearrangement of the definition of a radian: \text{radian} = \frac{s}{r}

  • Angular kinematic variables ($\theta, \omega, \alpha) must be in radians for the equations to work correctly.

  • Radians are technically unitless, while degrees are a unit. This is why radians are required to maintain correct unit consistency.

Relationship Between Linear and Angular Quantities

  • s = r \theta where:

    • s = arc length

    • r = radius

    • $\theta$ = angular displacement (in radians)

  • The angular kinematic variable must be in radians.

  • Radians is a unitless quantity. If using degrees, degrees are a unit so you'd end up with a meter degrees, which does not match what you want out of it.

Velocity on a Circular Path

  • Instantaneous velocity is tangent to the circular path.

  • v_t = r \omega where:

    • v_t = magnitude of the linear velocity (tangential)

    • $\omega$ = magnitude of angular velocity

    • r = radius

  • Use positive values for angular velocity and r. Only looking for the length of that arc length.

Linear Accelerations

  • Two perpendicular components:

    • Tangent to the path (a_t)

    • Perpendicular/normal to the path (a_n)

Tangential Acceleration

  • a_t = r \alpha where:

    • $\alpha$ = magnitude of angular acceleration

  • If angular velocity is constant (\alpha = 0), then a_t = 0.

Normal/Centripetal/Radial Acceleration

  • Always directed towards the center of the circle.

  • an = ac = a_r = r \omega^2

  • Alternative form: a_n = \frac{v^2}{r}, where v is the tangential velocity.

  • Finding the magnitude of linear acceleration is: \sqrt{at^2 + an^2}

  • As the radius increases, linear attributes also increase.

  • motion = normal/centripetal acceleration

Centripetal vs. Centrifugal Force

  • Centripetal Force:

    • A force inward is a thing that exists.

    • Causes centripetal acceleration.

  • Centrifugal Force:

    • Doesn't exist.

    • The phenomenon that we observe because of the lack of a centripetal force to keep it on that circular path.

  • If restraints are removed, the object will want to get further away from the center because the velocity is tangent to the path.

  • Centrifuge: works off of the lack of the centripetal force on that object that's suspended in there.

Biceps Curl Example

  • Analysis of an and at during a biceps curl.

  • an is proportional to \omega^2 and at is proportional to \alpha.

  • The forearm segment angle increases as we flex the joint.

  • Magnitude of angular velocity squared will be hill hill.

  • Magnitude of angular acceleration becomes a hill hill and a hill hill profile.

  • Angular profiles of the forearm segment are:

    • Angle: Increases during flexion, decreases during extension.

    • Angular Velocity \omega: Hill-valley profile.

    • Angular Acceleration \alpha: Hill-valley-valley-hill profile.

  • The equations for an and at:

    • a_n = r \omega^2

    • a_t = r \alpha

Analysis of \mathbf{a_n}

  • Since we only care about the magnitude:

    • Angular velocity, which had a hill-valley profile, becomes hill-hill after being squared. Larger velocity values become more accentuated with squaring.

    • a_n has a hill-hill shape. As the angular velocity increases, more centripetal force is needed.

Analysis of \mathbf{a_t}

  • The angular acceleration ($\alpha) profile, which is hill-valley-valley-hill, becomes hill-hill-hill-hill after taking the magnitude.

Halfway Point of Up Phase

  • Vertical acceleration is zero.

  • Tangential acceleration is zero.

  • Horizontal acceleration is large and negative.

  • Normal/centripetal/radial acceleration is large.

Effect of Repetition Speed

  • Changing repetition speed alters \omega by changing \Delta t.

  • Acceleration has a squared effect.

  • Forearm and hand length have linear effects, not squared effects like changes in time.

Rolling Ball Example

  • Relationship between linear and angular velocity:

    • v{cm} = r \omega or \omega = \frac{v{cm}}{r}

  • At any instant, the point on the ball touching the ground has zero linear velocity if the ball is not slipping.

Torque

  • Main construct of angular kinetics.

  • Analogous to force in linear kinetics.

  • Capital T is the variable, and it is a vector quantity.

Definition

  • The rotational analog to force.

  • The effect of a force that tends to cause a change in a body's state of angular position or motion.

  • Units: Newton-meter (N·m) or foot-pound (ft·lb) in the English system

    • 1 N·m = 0.736 ft·lb

  • Biomechanists prefer the term "moment" or "moment of force".

Calculation

  • Torque is the product of the magnitude of a force and its moment arm.

  • A vector about an axis perpendicular to the plane of the force and moment arm.

Convention

  • Counterclockwise rotation: positive torque

  • Clockwise rotation: negative torque

  • Magnitude of torque = (magnitude of force) * (length of moment arm).

  • The moment arm is the perpendicular distance from the line of action of the force to the axis of rotation.

Subscripts

  • Use a subscript on T to connect it with the force (e.g., T_F).

  • Sometimes a subscript is associated with the axis of rotation (e.g., T_A).

Example: Free Body Diagram of Forearm and Hand

  • Forces:

    • Bone-on-bone contact force

    • Muscle force (F_m)

    • Weight of forearm and hand

    • Force from barbell

Torque Calculations

  • Torque from muscle: Tm = |Fm| * r_m

  • Torque from forearm and hand (weight): Tw = -|W| * rw

  • Torque from force in hand: TH = -|FH| * r_H

  • Torque from bone on bone is zero because the line of action goes through the axis of rotation, so arm is 0.

  • These torques have different signs because some promote rotation in different directions.

Lever Arm vs. Moment Arm

  • Avoid the term "lever arm" because it has two definitions:

    • Synonymous with moment arm (perpendicular distance).

    • Straight-line distance from the axis to the point of force application.

  • Leverage: conveys the notion of increasing the moment arm.

  • The moment arm is the perpendicular distance from the line of action of the force to the axis of rotation.

Torque as a Vector Cross Product

  • Avoid this approach.

  • The way Torque is a vector cross product is T = F \times \Delta p

  • T = F \times \Delta P \sin(\theta)

  • $\Delta P$: A displacement to the point of application from the axis which is just ends up being the length.

  • But instead the preferred way is T = F \times R