Lecture 12, Revisiting Newton's Law - Torques Creating Rotations Flashcards
Newton’s First Law
the law of inertia - every object has a desire to remain in a static state (because it is easier, do not require any extra force or energy, keep doing it is what you are doing)
angular motion
- every body persists in its state of being at rest or rotating uniformly, except insofar as it is compelled to change its state by torque impressed (continue rotating or spinning at the exact same rate unless acted upon by a external torque that causes you to speed up or slow down or you are not rotating and persist doing that until a external causes you to start rotating)
- the system is made up of individual parts that can move and when we talk about rotations it is about how you move with respect to axis if rotation and not just you (have to know where it is): need to know how you are put together and where the axis of rotation
Calculating Rotational Inertia
- moment of inertia (rotational inertia) is the name given to rotational inertia, the resistance of any physical object to any change in its angular state of motion
- moment of inertia (I) = (mass) x (radius of gyration)2
- I = mk2
- units: kgm2
- the radius of gyration is the linear distance (measured in meters) between the axis of rotation to a point where the body mass is concentrated
- the moment of inertia is specific to a chosen axis of rotation
- when we quantifying inertia for an angular perspective we are still looking at how much stuff you are made up but also with respect to the axis of rotation (where is all the stuff centered with respect to the axis of rotation)
- anytime you hear moment it is associated with rotation
- how much stuff you are made off and where it is centered on a system (measure a distance with respect to axis of rotation)
- radius of gyration: distance between axis of rotation to where the masses are centered on point of interest/system (specific part) - similar to radius of rotation
Principal Moment of Inertia (planes)
the principal moment of inertia examines the amount of resistance to change relative to one of the principal axes of the body
how difficult it is to move the body in a particular direction
- what sort of rotational movements occur in each plane?
◦ inertia is low in the transverse plane (least amount of torque to rotate) - masses tend to be fairly close to the longitudinal axis meaning it is fairly easy to say no where it is harder to say yes - closest to axis of rotation
◦ moderate inertia in the sagittal plane - it hardest to be indecisive (the distribution around the axis is greater)
◦ inertia is highest in the frontal plane (most amount of torque to rotate head) - wide spread masses
my notes:
- k represents how wide apart you are and how close the masses are to the axis of rotation
- 3 planes you can move about (transverse, frontal and sagittal) where the mass are spread about differently in each which can make it harder or easier in terms of requiring more torque
- in certain planes, some masses are spread more which can increase difficulty to move the body
- the distribution around axis of rotation can change
- mass does not change but what changes is how close or far away the masses are to axis of rotation (the distribution around the axis of rotation)
- need to generate torque to overcome inertia
Manipulating Intertia
- to move, we must overcome inertia
- rotational inertia is affected by the distribution of mass in a system
- moment of inertia is the angular counterpart to mass
◦ unlike mass, the moment of inertia is not a fixed concept - rotational inertia is dependent upon the distribution of body masses
◦ we can manipulate the radius of gyration to alter the moment of inertia
my notes:
- we are also dealing with distance and not just mass (can change around distance which can change the amount of resistance you face when moving - if you can change the moment of inertia you can change the amount of resistance you face)
- the mass of your arm (does not change) and where they are centered with respect to axis of rotation are two things that determine how easily we can overcome resistance and inertia
- by shortening radius of gyration (everything bends when we run) - shortening the distances between where the masses are centered and axis of rotation (decreasing rotational inertia) - it makes it easier it to move as you can put more energy to running instead of swinging the limbs - can manipulate your rotational inertia
- the mass never changes but how they are distributed around axis of rotation does
Changing Inertia - What is the best method of carrying the item?
- as k increases your moment of inertia increases you have to do more to change state of motion
- as k decreases your moment of inertia decreases meaning you have to do less to move the object
- in order to pick up an object you need to overcome its inertia and you are changing the objects distances
- when we interact with objects where we grab an object matters, grabbing closer to centre of masses and axis of rotation is easier (easier to manipulate an object this way)
- will be better to use technique 2 as you are picking it up closer to where the masses are centered - grabbing on to the heavier end is easier to manipulate because you have to do less torque in order to hold on to the object
The Conservation of Angular Momentum
- when angular momentum is conserved, there is a tradeoff between moment of inertia and angular velocity
H = lω or H = mk2ω
H is constant and m is constant
but k2↑so ω must↓ (angular velocity decreases if you increase your radius of gyration - total angular momentum needs to remain the same)
but k2↓ must so ω must↑(angular velocity increases as you shrink as your bring your arms in) - if you spread you arms out you are going to slow down
- once you release yourself in the air if there is no external forces acting on a system whatever your momentum is in one instance will be the momentum in another
- as you spread your arms out you are increasing k (the distribution of you masses) and when you bring your arms closers you are shrinking the distribution of your masses - can play around with our radius of gyration
- when skaters want to go faster they bring themselves as tight as they can
Angular Momentum and Newton’ First Law
- when an object rotates, it has angular momentum
◦ angular momentum is the quantity of motion
◦ angular momentum (H) = moment of inertia x angular velocity
◦ H = Iω
◦ H = mk2ω
◦ units: kg m2/s - according to Newton’s First Law, in the absence of an external torque every system remains in a constant rotational state
- so, the angular momentum in one instance should be the same as the angular momentum in another instance (H1 = H2)
- conservation of angular momentum - whatever your angular momentum was in one instance will be the same in another a little later on (exact same concept as linear)
my notes: - the moment you move you have momentum and the ability to rotate (effect another system)
- angular momentum is the effect a rotating system has on another
- you substitute mass with moment of inertia for angular momentum (quiz question)
- radius of gyration is a linear measurement (in metres) so everything must be measured in radians because general motion (blank per seconds)
The Conservation of Angular Momentum (diver)
- in the absence of an external torque every system remains in a constant rotational state
- although total body angular momentum is conserved, the diver can alter their orientation in space
- independently moving the arms can shift the axis of rotation
- the diver will start to twist and spin in the air
- the direction of the spin depends upon which arm moves
- the diver can change the rate of spin but also their position in the air in terms of the direction in which they move through the air with your legs and arms one way or the other
- no extra force just changing the way the forces act on you
The Conservation of Angular Momentum - Asymmetrical Movements (Volleyball)
COMPENSATORY movement
- kicking the legs forward will cause the body to rotate backwards
- there are doing two different things (arm is doing completely different things from legs)
counter-rotation
- keeps the body’s position stationary
INITIAL movement
- swinging the arm at the shoulder joint will cause the whole body to
rotate forward
- the movement of arm with leg keeps everything level
my notes:
- postural reflexes are built within us to try to control the body position - if you get bumped you do not want to fall forward so you swing your arms a bunch
- by lifting your arms up it changes your centre of mass where gravity is going to pull you in a backwards direction if you are not careful (as the centre of mass moves towards the back part of your body) - gravity pulls you down if there more force acting backwards but you can counter it by attempting to move forwards
- if you are going to be up in the air you need to do it with control - volleyball players as they move their arm forward at the same time they will kick their legs forward (compensatory action) but with the legs in isolation it changes your center of masses wanting you to fall backwards
- so in order to keep everything level you do a asymmetrical action to keep everything level
Calculating Angular Momentum (total)
since moment of inertia is affected by BOTH body and segment orientation, momentum must consider the entire system
- local term - examining the momentum of an individual segment of the system (individual body parts - as you go to move through the air what is moving, for example) - every local term that is moving has an effect on the remote term
- remote term - examining the momentum of the entire system (where is the position of the overall system)
calculating the total-body angular momentum for a particular movement
- need to look at all parts of someone when calculating momentum
- in order to have an effect you need to take into account two things: the mass and velocity of the system
Newton’s Second Law
the law of momentum: creating a change in the state of motion (dynamic motion)
angular motion
- the rate of change on angular momentum of a body is proportional to the torque causing it and the change takes place in the direction in which the torque acts
- where you either create or stop rotation
- it is not about calculating momentum it is about changing momentum
Angular Impulse and Newton’s Second Law
a change in angular momentum is produced by an angular impulse
- angular impulse = torque x time
- angular impulse = Tt
- units: Nms
- can we write the formula another way?
◦ angular impulse = Tt
◦ angular impulse = △angular
momentum
◦ △angular momentum = H2 –
H1
◦ △angular momentum = Iω2 –
Iω1
◦ Tt = Iω2 – Iω1
- impulse creates change in momentum
- whatever your angular momentum in one instance will be different in another as we are changing the state of motion
Newton’s Second Law (acceleration)
the law of acceleration: creating a change in the state of motion (dynamic motion) - by either changing momentum in changing velocity or if you change velocity you are also changing acceleration
angular motion
- torque applied to a body causes an angular acceleration of that body of a magnitude proportional to the torque, in the direction of the torque, and inversely proportional to the body’s moment of inertia
- a torque applied to a moment of inertia creates angular acceleration - desire to change state or be in a dynamic state
Newton’s Second Law Re-Examined
Why is Newton’s Second Law referred to as the Law of Momentum AND the Law of Acceleration?
the law of momentum: Tt = Iω2 – Iω1
divide both sides by t: T = Iω2 – Iω1 / t
T = I (ω2 – ω1) / t
T = Iα
a change in velocity is? the law of acceleration
the torque applied over a period of time is going to change acceleration (you change angular velocity as mass and position are staying the same) - torque dominates this change
Newton’s Third Law
the law of reaction: for every action there is an opposite and equal reaction
linear motion
- if a body impinges upon another, and by its force changes the motion of the other, that body also will undergo an equal change, in its own motion, toward the contrary part
angular motion
- if a body impinges upon another, and by its torque changes the motion of the other, that body also will undergo an equal change, in its own motion, toward the contrary part
- you spin about the ground the ground spins back around equally
