Torque and Angular Kinetics set 8A

Review of Problematic Questions

• Question 1: Runner's Free Body Diagram

  • Scenario: Known runner mass, ground reaction force, acceleration center mass, and solution for horizontal air resistance.

  • Task: Determine FA (air resistance) in standard polar notation.

  • Key points:

    • A negative force indicates it acts opposite to the drawn direction.

    • Polar notation always reports the magnitude (positive number).

    • If FA acts to the right (positive X direction), the angle is 0 degrees.

    • Correct answer: 50 Newtons at 0 degrees.

• Question 2: Vertical Component of Force

  • Scenario: 20 Newton average force at a polar angle of 40 degrees.

  • Task: Determine the vertical component.

  • Key points:

    • Vertical component = $20 \sin(40)$.

    • Also equivalent to $20 \cos(50)$.

    • Correct answer: More than one of the above.

• Question 3: Biceps Curl Resultant Force

  • Scenario: Halfway point of the down phase of a biceps curl, forearm and hand horizontal.

  • Task: Determine the quadrant of the resultant instantaneous force applied by the hand to the barbell.

  • Key points:

    • Horizontal force is large and negative (pulling the barbell).

    • Vertical force equals the weight of the object (upwards).

    • Resultant vector acts up and to the left (second quadrant).

    • The force from the hand $F<em>x = m a</em>x$

Torque Defined

• Elbow as the axis of rotation.

• Free body diagram:

  • Force at the hand (action-reaction from barbell).

  • Weight of forearm and hand acting through the center mass.

  • Muscle force (simplified singular muscle).

  • Force at the bone.

• Torque = Magnitude of force × Length of the moment arm (perpendicular distance from the line of action of force to the axis of rotation).

• Determine the direction of the turning (clockwise or counterclockwise) and add a $+$ or $-$ sign.

Torque as a Vector Cross Product

• Torque = Magnitude of force × Magnitude of vector from axis of rotation to point of application × Sine of the angle between the two vectors.

• Formula: $\tau = |r| |F| \sin(\theta)$

• Delta P sine theta equals the moment arm.

• This approach relies on polar setup.

• Cartesian components are often more practical for calculating torque.

Cartesian Components of Torque

• Resolve muscle force into vertical and horizontal components ($F<em>y$ and $F</em>x$).

• Vertical force has a horizontal moment arm, and vice versa.

• Torque made by components sums to the resultant torque.

• Torque from the muscle can be calculated using the formula: $\tau<em>m = r</em>x F<em>y + r</em>y F_x$

• FX component tends to cause a clockwise (negative) rotation.

• FY component tends to cause a counterclockwise (positive) rotation.

• When looking at resultant forces, each must be assessed independently for its rotation direction.

• X and Y components will sum together for the same resultant.

• Advantages of using Cartesian components:

  • Forces are typically broken down into X and Y components.

  • Geometry is typically given as coordinate information.

• Moment arms should have subscripts that match the force ($m<em>x$ with $F</em>x$, $m<em>y$ with $F</em>y$). A horizontal force has a moment arm vertical, but it goes along with X force and the X links it to the force.

Key Considerations for Torque

• X and Y components can have different senses of direction.

• A moment arm with a Y subscript may be a horizontal distance, and vice versa.

• Vertical-only vector: only a Y component.

• Horizontal-only force: No Y component.

Biceps Curl Analysis with Torque

• Vertical force pushes against the hand.

• As we flex and extend, what happens to the moment arm?

• Starting position: horizontal distance is the moment arm.

• Flexed position (mid-range): horizontal distance out to that spot.

• Continued flexion: horizontal distance from elbow to hand.

• Moment arm shape: Hill-shaped curve (small to large to small).

Moment Arm with Horizontal Force Component:

• Horizontal force exists, moment arm becomes a vertical distance.

• At the beginning, the moment arm is the vertical distance up to the elbow joint.

• As weight rises, the vertical distance decreases to zero mid-rep.

• As weight further rises: the distances get larger again.

• Moment Arm Characteristics:

  • Vertical Force: Small to large to small.

    • horizintal distances changes throught out felxion/extension

    • hill shapes cufve during the up phase

  • Horizontal Force: Large to small to large.

    • vertical distance

Impact of Forearm Length on Moment Arms

• Longer forearm: increases moment arms.

• Vertical force: increases arc; hill-shaped curve becomes higher.

• Linear increase due to sine/cosine angles remaining constant; only forearm length changes.

Influence of Speed and Mass on Moment Arms

• Repetition speed (faster or slower) does not change moment arms, which are purely geometric.

• Mass also has no influence.

Torque and Quasi-Static Biceps Curl

• Quasi-static: accelerations converge to zero.

• Horizontal force converges to zero.

• Vertical force approximates the weight of the barbell ($-W$ - minus is for direction purposes).

• Torque at the hand: $\tau = -W \times r_y$

• Shape is a hill and hill.

• Vertical force - horizontal moment arm

• Shape of profile: Valley in a valley.

Newton's Laws Angular Application Analog

• First Law (Law of Inertia):

  • Linear: Summation of external forces equals zero.

  • Angular: Summation of external torques on object equals zero.

• Second Law (Law of Acceleration):

  • Linear: Summation of forces equals mass times acceleration ($F = ma$).

    • $m$: mass - linear inertia

  • Angular: Summation of torques equals moment of inertia times angular acceleration ($\tau = I \alpha$).

  • Alpha = angular acceleration.

  • I = Moment of inertia (angular analog of mass).

    • Variable I is angular inertia

    • a capital $O$ placeholder will be used to remind us that it is about the origin.

• Third Law (Law of Action-Reaction):

  • Linear: For every force, there is an equal and opposite force.

  • Angular: For every torque, there is an equal and opposite torque somewhere on contrary body.

Understanding Moment of Inertia (I)

Mass Moment of Inertia:

Technically there are different kinds, mass is in parentheses usually, we will come across it later in the semester but, most of the time, people don't bother to say the word mass and just call it MOI.

• Angular analog to mass.

• Units: Kilogram meters squared ($kg ". m^2$).

• Quantifies object's resistance to changes in rotary motion.

• Scalar quantity.

• Based on the distribution of mass about the axis of rotation.

• Different moment of inertia value for every different axis of rotation of an object.

Examples

Baseball Bat Example:

Discussing this due to the idea that every axis of rotation may have a different MOI and one of the reasons why we need to have the subscript there.

• Swinging the bat normally: mass far away from the axis.

• Grabbing the barrel of the bat: easier to swing because mass is closer to the axis.

• Smallest moment of inertia at the center of mass.

• Choking up on the bat: reduces the moment of inertia.

• Mass = 1, but infinite MOI

The Actual Equation:

• $I = \sum<em>{i=1}^{N} m</em>i r_i^2$

• N = the number of sub elements think that your object is made of.

• m = mass of each particle and where they're located.

• R = straight line distance from the axis to the mass.

  • Got to think about accounting for every little mass that makes up your object and appropriately at a density to get the geometry taken care of to get our accurate distribution like that.

Equation Summary:

• Key thing is to only square the $R$ value.

  • For the generic potato:

  • $I = m<em>1 r</em>1^2 + m<em>2 r</em>2^2 + m<em>3 r</em>3^2 + m<em>4 r</em>4^2$

  • Small R values pack the mass real tight and get a small MOI.

Body Segments: Simplified Forearm Example

• Simplified because it is larger to the proximal end, more mass towards the proximal end. More mass towards the center mass than towards the distal end towards the wrist

• Two masses $m<em>1$ and $m</em>2$ as a simplified distribution to find the MOI.

• $m<em>1 = 2 * m</em>2$.

• 30 cm long with 10 cm in between masses and each end.

• Looking to rotate about the proximal end (Elbow).

  • Distances:

    • .1 and .2.

  • Proximal end MOI:

    • $I = m<em>1 r</em>1^2 + m<em>2 r</em>2^2$

    • Calculate

    • .03. $kg * m^2$

• Think about the numbers here of the R:

  • Rotating about wrist joint

    • $I$- distal end = m1$r</em>1^2 + m<em>2$r2^2

    • Calculate.

    • 0.45 $kg * m^2$

  • Key point = the same object, two different axes, one being 50% larger than the other.
    Not a whole lot different than the bat.

So Quick: to see that it is not a huge change: just a smaller mass moving around and moving the mass around to see that it increases. So a big piece comes from this moving it down there

• Our body is actually built this way with upper arms/forearms following this pattern. The design of that is great because you can always rotate more easily about the center mass.
  So that a lot of the segments look like that. Not only does our forearm look like that, where we've got the center of mass is gonna be closer to the proximal end. So it's gonna have a much smaller moment of inertia about the proximal end than if you rotate it about the distal end. But our upper arms are also distributed that way. Got a lot more mass up here than we do closer to the elbow.
  *Designers new something:
  Who ever played a role in designing our bodies may have known something about moments of inertia in terms of having the muscle bellies packed closer to the proximal side of the joints, make them easier to rotate than if the masses were further away.

Dynamic and Static Benefit:

• Dynamic - moving the moments of inertia closer to the joint can accelerate that area.

• Static = holding the mass there.

• Dynamic benefit: reducing I means more torque goes into angular acceleration.

• Static benefit: less muscle torque needed to hold a limb in position.

• Double Bonus - for design and having masses closer to the proximal ends.