Revisiting Newton’s Law: Torques Creating Rotations

What is torque?

Torque is a moment

It is a force that has a tendency to rotate an object about a fixed axis

Angular: Newton’s 1st Law

Rotating uniformly — not moving OR spinning around in the same direction at the same rate, unless external torque is present

What is moment of inertia?

Name given to rotational inertia, the resistance of any physical object to change it’s angular state of motion

Instead of quantifying mass for inertia, we use mass and axis of rotation

Moment of inertia = Rotational inertia = I

What is radius of gyration?

Linear distance from where masses are centered on system and axis of rotation

• We want to know where the position of COG is

• Symbol for radius of gyr = k 

What is the relationship btwn the distrbution of mass and radius of gyration, and subsequently moment of inertia?

When distribution of mass increases, it increases radius of gyration (k), which increases moment of inertia (I), which increases your your resistance to motion and reduces your angular acceleration

e.g. skater putting their arms out wide reduces their angular velocity

What is the “principle moment of inertia” ?

The moment of inertia is specific to a chosen axis of rotation since the distribution of masses is diff when doing yaw (no), pitch (yes), and roll

Describe the degree of inertia in the: transverse plane

Low inertia

Least amount of torque to create rotational movement in the transverse horizontal plane (yaw) bc resistance is the lowest

• Distribution of masses is tightest to the axis of rotation

Describe the degree of inertia in the: saggital plane

Moderate inertia

Moderate amount of torque to create pitch (yes) as distribution of masses isn’t as tight or far.

Describe the degree of inertia in the: frontal plane

Highest inertia

High amount of torque to roll (lateral neck) movement since distribution of masses is the farthest from axis of rotation

What happens when you reposition the body segments such that the mass moves farther from axis of rot

Increases radius of gyration (k) → increases moment of inertia (I) → increases resistance to rotation/lowers angular acceleration

Conversely, repositioning body segments close to the axis of rotations lowers I, making rotation easier/faster

What is mass equivalent to in angular kinetics?

Moment of inertia is the angular counterpart to mass

Increasing mass (I) → Increases inertia → increases torque demands to produce rotational change

Is moment of inertia (I) a fixed concept like mass?

No. You can manipulate the dist. of mass (aka radius of gyration), while keeping the total mass the same, to change the rotational inertia and torque demands for rotation

Rotational inertia is depended on what?

Rotational inertia is depended on the distribution of masses

• We can manipulate the radius of gyration to alter the moment of inertia

What is the best method to carry a stanchion?

For a stanchion/chair, the masses are distribution closer to the heavy end. 
If we keep the axis of rotation (our arm/shoulder) close to the COG (where the masses center), radius of gyr (k) would be lower which would decreases moment of inertia

The opposite way would create a longer k causing I to be higher and hence harder.

What is angular momentum? UoM?

The quantity of rotational motion

• A rotating object carries angular momentum, so when it interacts with something else, part of that rotation can be transferred

Angular momentum (H) = moment of inertia (I) x angular velocity (w)

H = Iw

• H = mk² w

UoM: kg.m² /s

What is conservation of angular momentum?

Newton’s first law of rotation is the Conservation of Angular Momentum

In the absence of external torque, it’s angular momentum remains constant 

• H1 = H2 or Iw_{1 =} Iw_{2 or} mk²w_{1 =} mk²w^₂

• Even if you change I (tuck or spread), angular velocity automatically changes accordingly which means H will remain constant

• Only an external torque can change the amount of angular momentum. Otherwise you're only changing rotational inertia/resistance

Describe the tradeoff between I and w

When there is conservation of angular momentum, there is a tradeoff between I and w. 

H = mk² w, where H and m are constant

So when k increases, w must decrease v/v

How is conservation of angular momentum used to affect angular movements?

In the absence of external torque, the angular momentum stays the same

But you can shift the axis of rotation/COG/dist of masses to: 

1. Cause rotation in one direction or another

2. Cause slow or fast rotation

Apply and explain the conservation of momentum in terms of asymmetrical movements (initial movement, compensatory movement)

When volleyball player swings arm forward (initial movement), it creates angular momentum in the forward direction

Because there is no external torque, total H needs to remain the same so H1 = H2

To prevent the body from falling forward, the player performs a compensatory movement by kicking the legs which causes the body to fall backwards

Because the initial movement (arms) cause body to fall forwards, and compensatory movement (kicking legs) cause body to fall backwards, the counter-rotation causes the body’s position to remain stationary.

Long jump kinetics

The arms and legs rotate to prevent the trunk+head from rotating backwards

Since the moment of inertia for trunk+head is larger than the limbs, the ang velocity of the limbs (arms + legs) must be larger in order to stabilize the position of the body in the air

Newton’s Second Law of Rotation

Law of Angular Momentum

The rate of change of angular momentum of a body is:

1. proportional to the torque causing it

2. the change takes place in the direction in which the torque is acting

What causes a change in angular momentum? UoM?

A change in angular moment is produced by an angular impulse

Angular Impulse = torque x time

Angular impulse = Tt

UoM: Nms

Rewrite angular impluse (Tt)

Tt = H2 -H1 (since law of ang mom is not about calculating mom, it’s about the change in mom)

Tt = Iw₂ - Iw₁

What is external torque doing to the system to change angular momentum?

External torque is NOT changing the moment of inertia (mass or radius of gyration) it is only changing/affecting the angular velocity over a time period

Why is the Newton’s 2nd Law referred to as the Law of Momentum AND the Law of Acceleration

Law of Momentum isn’t about calculating momentum, it’s about change in momentum:

Tt = Iw2 - Iw1

What creates the change in H is the torque, which does not affect the I, only the angular velocity over a period of time. 
Change in velocity = acceleration 

Tt = Iw2 - Iw1 ← Divide both sides by t

T = I(w2-w1)/t

Torque (T) = I⍺

Newton’s 3rd Law of Rotation

If you apply torque on a system and change it’s rotation, it applies an equal in magnitude and opposite in direction torque back on to you

State the angular equivalents:

Study of forces

Force (N)

Mass (kg)

Momentum (Nm/s) 

Impulse (Ns)

• Study of torques

• Torque

  • Nm

  • Force x moment arm

• Moment of Inertia (mass x radius of gyration) 

  • kgm²

• Angular momentum

  • individual segments + whole system

  • kgm²/s

• Angular Impulse 

  • Nms

Explain what a diver is doing when she extends prior to entering the water at the completion of a dive.

Diver extends to distribute her masses away from the axis of rotation → increase radius of gyration (k) → increases moment of inertia (I)

To conserve the angular momentum, angular velocity decreases. This allows a clean entry to water and make less of a splash 

Why do volleyball players swing their legs forward when spiking a ball with their shoulder?

Kicking the legs is a compensatory movement that pushes the body backwards causing a counter-rotation to the forward motion created by the spike with the shoulder, allowing them to land in the same position that they jumped
Otherwise, the initial movement of spiking the ball shifts the COG up and forward which would cause them to fall forward

What is the linear equivalent to moment of inertia?

Mass, which is a fixed number.
I isn’t

What are the units for angular momentum?

H (ang mom) = I (moment of inertia) x w (angular velocity)

OR 

H = mk²w

UoM = kgm²/s

What are the units for angular velocity?

rad/s

Which principal axis of rotation has the greatest moment of inertia in the anatomical position?

The antero-posterior axis of rotation (frontal plane/roll) has the greatest moment of inertia