Linear Kinetics Pt. 1

LINEAR KINETICS

Overview of Kinetics

Kinetics involves understanding the forces that act on a system and what causes motion.
Motion can be categorized as linear when it occurs in a straight line, or translatory motion.

Measuring Kinetic Variables

Measurement of kinetic variables can be performed using various equipment such as:
- Force platforms
- Plates
- Walkways
- Instrumented treadmills

Forces

Definition: A force is a push or a pull acting on a body that can produce, stop, accelerate, or change the direction of motion.
The acceleration of the object may change or be prevented from changing based on the forces applied.
Unit of Measurement:
- Newton (N), where 1N = 1kg imes 1m/s^{2}

Properties of Forces

Each force has four main characteristics:
1. Magnitude
2. Direction
3. Point of Application
4. Line of Action
Point of Application: It refers to the specific point at which the force is applied. This determines if the resulting motion is linear or angular.
Line of Action: The straight line in the direction of the applied force; assuming a single acceleration of the object along it.
The application of a force results in the acceleration of a body’s mass.

NEWTON'S LAWS

Newton outlined three fundamental laws of motion that govern dynamics:
1. Law of Inertia
2. Law of Acceleration
3. Law of Reaction

NEWTON’S FIRST LAW: Law of Inertia

A body maintains its state of rest or constant velocity unless compelled by a net external force to change that state.
Key points include:
- A motionless object remains stationary without a net force acting on it.
- A body moving with constant speed continues its motion unless a net force acts upon it.
- Example: A skater gliding on ice continues with the same speed/direction until acted upon by friction or air resistance.
This can be mathematically expressed as:
- If ext{ΣF} = 0 then ext{Δv} = 0
- Here, ext{ΣF} represents the net external force affecting the object.
To overcome inertia, a net force larger than the inertia of the object is needed to initiate acceleration.

Inertia

Inertia is defined as the resistance to action or change, reflecting a body’s tendency to remain in its current motion state.
It is proportional to the mass of the body without any units of measurement.
Greater mass necessitates larger external forces to overcome inertia and induce acceleration.

NEWTON’S SECOND LAW: Law of Acceleration

According to this law, a force applied to an object results in an acceleration that is:
- Proportional to the magnitude of the force
- In the direction of the force
- Inversely proportional to the mass of the object
Mathematically:
F = ma
This implies:
- For a thrown or kicked ball, it travels in the direction indicated by the line of action of the applied force.
- The greater the force applied, the greater the resultant speed.
- When multiple forces act in opposition, they need to be considered for net effect. If forces are balanced (net force is zero), there is no acceleration (per the first law).
- Conversely, if net force produces acceleration, the object will move in a straight line along the line of action of the net force.

Mass

Definition: Mass is the quantity of matter in an object.
Measurement unit: kilograms (kg)
It is considered a scalar quantity and constant irrespective of location (Earth or Moon).

NEWTON’S THIRD LAW: Law of Reaction

It states that for every action, there is an equal and opposite reaction.
In terms of forces: When one body exerts a force on another, the second body exerts a reaction force that is equal in magnitude and opposite in direction.
Mathematically represented as:
ext{ΣF}{A on B} = - ext{ΣF}{B on A}
Example in human movement: A jumper applies a downward force against the ground, which results in an upward force (reaction) that helps in jumping and stabilizing the landing.
Ground reaction forces (GRFs) are vital in analyzing gait patterns and motion dynamics.

PRACTICE PROBLEM

Scenario: A 90 kg ice hockey player collides with an 80 kg player, exerting a force of 450 N on the latter.
- According to Newton's Third Law, the force exerted by the second player on the first is equally 450 N but in the opposite direction.
- Therefore, F_2 = -450N.

SOCRATIVE PRACTICE QUESTIONS

What biomechanics technology can be used to measure kinetic variables?
Identify the proper units for force.
Indicate which of the following is not one of Newton's 3 laws of motion.
True or False: The Law of Reaction is expressed as ext{ΣF}{A on B} = ext{ΣF}{B on A}.
Which block has the highest inertia?

VECTOR COMPOSITION

Definition: The composition of two or more vectors involves adding their magnitudes to determine a resultant vector.

Same Direction Vectors

When two or more vectors point in the same direction, the resultant vector's magnitude equals the sum of the individual magnitudes.

Opposite Direction Vectors

When vectors oppose each other, the resultant vector's magnitude equals the difference of magnitudes in the direction of the longer vector.

Non-collinear Vectors

For vectors not aligned either in the same or opposite direction, the resultant vector is determined by the tip-to-tail method - positioning the tail of the second vector to the tip of the first.
The resultant vector connects the tip and tail of the arranged vectors.

Resolution of Force Vectors

To analyze forces thoroughly, vectors can be resolved into perpendicular components or combined into resultant vectors representing the net effect of all forces in the system.
Any system of forces residing within the same plane is termed coplanar, and if they converge at a single point, they are considered concurrent; such forces can be replaced by a single resultant vector.

Example Problem

Consider Vector A with a length of 10 at angle 45:
- Vertical Component (y):
  y_A = 10 imes ext{sin}(45) = 10 imes 0.7071 = 7.07
- Horizontal Component (x):
  x_A = 10 imes ext{cos}(45) = 10 imes 0.7071 = 7.07
Vector B has length 6 at angle 0°:
- Vertical Component (y):
  y_B = 6 imes ext{sin}(0) = 0
- Horizontal Component (x):
  x_B = 6 imes ext{cos}(0) = 6.00
Calculate the resultant vector's components and their magnitudes.

Resultant Magnitude Calculation

Use the Pythagorean theorem to find the resultant or hypotenuse:
c^2 = a^2 + b^2
For the above example: 16.40^{2} + 2.57^{2} = c^{2}
- Resulting in c = ext{√275.57} = 16.60
Subsequently, find the resultant angle:
ext{angle} = an^{-1}igg(rac{2.57}{16.40}igg)
ightarrow ext{approximately } 8.91°

Example in Real Context

Application: A canoe navigating a river experiences forces from the current and the wind, necessitating vector composition to identify the net force directing the canoe.

QUIZ QUESTION

Quiz items throughout were focused on understanding force vectors, resultant calculations, and applying Newton's laws.

Types of Forces

Forces can be categorized generally as contact and noncontact forces.

Contact Forces

Definition: Forces exerted through direct contact between two objects.
Examples include:
- A bat hitting a baseball
- A foot striking the ground

Noncontact Forces

Definition: Forces acting at a distance, where objects are not in direct contact.
Key example:
- Gravitational force pulling objects toward the Earth.
Law of Gravitation: The gravitational force is inversely proportional to the square of the distance between two masses and directly proportional to the product of the masses involved:
F = rac{G m_{1} m_{2}}{r^{2}} where G is the universal gravitational constant.

Weight as a Noncontact Force

Weight signifies the amount of gravitational force acting on a body, calculated using:
Wt = mg
Units: Weight is measured in Newtons (N) or pounds (lbs), not in kilograms (kg).
The gravitational acceleration (g) is roughly -9.81 \, m/s^{2}, indicating the downward direction towards the Earth.
Weight has magnitude, direction, and a common point of application at the center of gravity.

Center of Gravity

The center of gravity delineates the point around which weight is symmetrically balanced across a body in any positional orientation.
The location of the center of mass is pivotal in motion analyses as it influences how the body interacts with applied external forces.
The center of gravity pertains only to vertical directionality due to gravity, while the center of mass is independent of directional orientation.

CONTACT FORCES

Contact forces arise when objects physically interact, crucial to human movement, examples include:
- Ground Reaction Force (GRF)
- Joint Reaction Force
- Friction
- Fluid Resistance
- Inertial Force
- Muscle Force
- Elastic Force

Ground Reaction Force (GRF)

The force exerted by a surface in response to exertion by an individual.
- GRF varies across different surfaces and terrains, influencing motion analysis.
- When an individual pushes against the ground, the surface pushes back with an equal and opposite force (Newton's Third Law).
- GRF components include:
- F_{z}: vertical (up-down)
- F_{y}: anteroposterior (forward-backward)
- F_{x}: mediolateral (side-side)
Prioritized analysis typically focuses on the vertical component for gait analysis, while anteroposterior and mediolateral components are considered shear components.

Joint Reaction Force

The net force acting across a joint.
Calculable based on kinematic and kinetic data plus anthropometric variables.
For example: Standing still, the thigh applies a downward force on the leg at the knee joint, countered by an upward force from the leg on the thigh, as muscle contraction provides bone-on-bone forces.

QUIZ QUESTIONS

Gravity is a contact force (True/False)
The point of origin of the weight vector is the ___________?
Calculate the mass of an object weighing 1200 N.
Determine the required force to lift a 70 kg barbell.
If a 4.12 N ball needs to accelerate at 20 \, m/s^{2} to reach a target height, find the vertical force required.

FRICTION AS A CONTACT FORCE

Definition: Friction is a force that counteracts the motion between two surfaces in contact.
It occurs opposite to the intended motion direction and must be overcome to initiate movement.
The frictional force (Ff) can be expressed as:
F_f = ext{μN}
Where µ is the coefficient of friction and N is the normal force (weight of the body).

Coefficient of Friction (CoF)

Definition: The coefficient of friction is dimensionless and illustrates the interaction degree between two contact surfaces.
A higher coefficient implies increased molecular interaction, leading to greater friction.
To initiate motion, sufficient force must overcome the maximum static friction force F_{s ext{MAX}}, represented by:
F_{s ext{MAX}} ext{ }(F_m) = ext{μ}_{s}N

Static vs Kinetic Friction

Static Friction (FsMAX) is the maximum friction force before movement, whereas Kinetic Friction (Fk) relates to the friction when objects are already in motion.
Kinetic friction can be calculated using:
F_k = ext{μ}_{k}N
Kinetic friction remains constant and is always less than maximum static friction.

Example Problem - Sled with a Boy

If a boy weighing 250 N sits on a sled weighing 200 N with static friction (μs) of 0.18 and kinetic friction (μk) of 0.15:
1. Force to start motion (Static):
  F_m = μ_sR = (0.18)(250 N + 200 N) = 81 N
2. Force to maintain motion (Kinetic):
  F_k = μ_kR = (0.15)(250 N + 200 N) = 67.5 N

FRICTION ALTERATIONS AND CONTEXT

Changes in the normal force affect friction magnitude; increasing weight raises friction, making it easier to initiate motion if N decreases.

Direction of Normal Force

When an object is pushed or pulled, the angle of applied force alters the normal force's vertical component, influencing the friction experienced.
Example: Pushing down on a desk increases its normal reaction force and therefore the friction force, while pulling up reduces its normal force and thus the friction encountered.

Practical Applications of Friction

Modifying coefficient values can enhance friction for various activities:
- Skaters prefer freshly frozen ice for lower friction.
- Golfers wear gloves to increase friction with the ball.
- Cleats provide better traction but may yield excessive friction in some situations, risking injury.

QUIZ QUESTIONS

Quiz statements focused on friction's characteristics, definitions, and calculations relevant to various scenarios.