Mechanics and Forces

Branches of Mechanics

• Statics deals with objects at rest or those moving at a constant velocity.
• Dynamics focuses on the mechanics of objects experiencing accelerated motion.
• Kinematics describes motion.
• Kinetics looks at forces, and how they cause changes in motion.

Force

• Force is defined as a push or pull that initiates, stops, accelerates, decelerates, or changes the direction of an object.
• The SI unit is the Newton (N).
• 1N represents the force needed to accelerate a 1kg mass at 1m/s².
• Forces possess a point of application, a direction, and a magnitude.
• Force = mass x acceleration (F = ma).

Internal and External Forces

• Internal forces act within an object and affect the motion of its components.
• External forces act on an object due to its interaction with the environment.

Internal Forces

• Tensile forces are pulling forces (e.g., muscle action).
• Comprehensive forces are pushing forces (e.g., bone compression).
• In the human body, internal forces can alter the motion of a body part but cannot change motion for the body’s center of mass.
• The motion of the center of mass requires an external application of force.

External Forces

• Non-contact forces act without any physical contact (e.g., gravity).
• Contact forces transpire when objects touch (e.g., friction).

Non-Contact Force: Weight

• Weight: the force of gravity exerted on an object.
• Weight = mass x acceleration due to gravity (W = mg).

Directional Convention

• Upward or to the right is positive (+ R).
• Downward or to the left is negative (+ L).
• For problem-solving: treat gravity as negative with an exception to solve for potential energy.

Weight, Mass, and Inertia

• Weight: the force of gravity acting on an object.
• Mass: the quantity of matter within an object.
• Inertia: a property of matter resisting changes in motion; Objects maintain their state of rest or uniform motion unless acted upon by an external force.
• Mass is a key determinant of inertia.
• Mass is the measure of inertia, used to determine weight (F = ma) and is constant.
• Weight depends on the force of gravity (-9.81, except in specific contexts).

Paired Forces

• Forces come in pairs where one object exerts a force on another (action), and the second object exerts an equal and opposite force on the first (reaction).
• Paired forces is governed by Newton's Third Law of Motion.

Contact Forces

• Normal contact force is perpendicular to the surfaces in contact, and it is also called reaction force.
• In a horizontal system: normal contact force, R, is equal to the the objects weight (+ value).
• Friction force is parallel to the contact between two surfaces.
• Friction is caused by the molecular interaction between contact surfaces.
• Static friction happens when to surfaces are not in motion relative to each other.
• Dynamic friction happens when two surfaces are in motion relative to each other.
• Limiting friction: is the maximal static friction before movement begins.

Friction Force Calculation

• Friction force is proportional to the reaction force.
• Static friction force (Fs) = s x R, where s is the coefficient of static friction (0-1) and R is the reaction force.
• Dynamic friction force (FD) = D x R, where D is the coefficient of dynamic friction and R is the reaction force.
• Static friction is greater than dynamic friction.
• Less force is required to keep something moving than to start it.

Coefficients of Friction

• Various materials have coefficents of friction, and can be solved for with: (FS) = S x R → S = FS / R.

Friction and Surface Area

• Friction force remains constant regardless of the surface area in contact.

Vector vs Scalar

• Vector quantities have magnitude (size) and direction (velocity, acceleration).
• Scalar quantities have magnitude only (time, mass, speed).
• Scalar quantities have numbers and size but NO direction.

Force as a Vector

• A force has a point of application with the tip of arrowhead.
• Direction is the orientation of arrowhead.
• Magnitude is the length of the arrow.
• Vectors graphically represent direction and magnitude.

Net Force

• An objects movement depends on the SUM of all external forces acting on it. (∑F).
• If the ∑F = 0 the object is at rest (constant velocity/zero acceleration) this is an external force at equilibrium.
• The object accelerates in the direction of the net force of ∑F ≠ 0

Vector Addition

• To sketch a resultant force vector, add components tip-to-tail.
• Net force on an object is the vector sum of all external forces.
• Numbers cannot be added because direction must be considered.
• Vector composition is the addition of 2 or more force vectors produce a resultant force.

Vector Addition for Co-Linear Forces

• Simplest Case: Forces on an object exist on same line of action (co-linear).
• Direction can be either the same or opposite.
• Choose one direction to represent a positive force; all forces are acting in this direction can be added.
• All forces acting in opposite direction are added as negative numbers.

Vector Addition for Concurrent Forces

• Another vector addition involves forces whose lines of action are at the same point.
• Concurrent forces directed horizontally and vertically require summing horizontal and vertical forces (seperately).
• Horizontal (Fx) and Vertical (FY) components will dicatate the resultant force.
• Law of trigonometry determines magnitude (Pythagorean theorem) and direction (tan ratio) of the resultant force.

Static Equilibrium

• With an object at rest, the net force equals zero and the object is in static equilibrium.
• F = 0 in static equilibrium.
• ∑Fx = 0, ∑Fy = 0 allows one to solve for unknown forces.

Free-Body Diagram

• A mechanical representation (drawing) of the object (body) with all external force arrows (no internal forces shown).
• Arrows indicate the point of application, direction, and size, in the simplest form.
• Use a square, circle, stick figure, etc to define your object

Constructing a Free-Body Diagram

• Select a body/body segment with a simple drawing.
• Draw all known external forces at the point of application.
• Weight is a downward arrow at the center of gravity.
• Draw all unknown external forces at their points of application.
• Identify all unknowns to be included in the drawing.

Steps to Solve Mechanics Problems

• Draw a free-body diagram of the object.
• Draw an axis system to define positive directions, Y/X axes, or axes in degrees (0°-270°).
• The equation(s) of motion must apply to the problem, and be expanded using the information from the free-body diagram then solve for unknown(s).
• Final answers must use: the direction/sign (+/-); level of accuracy (1-2 decimals); appropriate units.

Vector Composition Basic Scenarios

• Vectors are co-linear- can be added- direction matters
• Vectors are horizontal and vertical (90° to each other).
• Magnitude is found by The Pythagorean.
• Direction is tangent ratio.
• Vectors on an Angle: angled vectors are split to horizonal and vertical conponents by vector resolution for one + triangles).

Motion

• A change of position with respect to space and time.
• An object in motion moves in a given space and the movement requires a given amount of time (linear, angular, general).

Linear Motion

• All points on an object move in the same distance, in the same direction, and at the same time.
• Recilinear transtlation occurs along a straight line.
• Curvilinear translation occurs along a curved path or arc.

Angular Motion

• All points on an object move in a circular path about the same fixed axis (elbow flection/extension).

Motion: General

• A combination of linear and angular movements walking/running).
• Linear and angular motions are analyzed separately when possible and different mechanical laws govern.

Characterizing Linear Kinematics

• Position
• Distance
• Displacement
• Speed
• Velocity
• Acceleration

Position

• The defintion is location given with a fixed reference point described using Cartesian coordinates.

Distance Travelled

• Length of path from object from start to end position.

Displacement

• The straight-line distance in a specific direction from the start to end position.
• Vector quantity like force that has arrows and resultant vector is found using trig.

Distance Travelled vs Displacement

• Numerically equivalent for 100m but drastically different in a 400m, 800m, 10k race.

Speed

• Rate of motion in m/s.
• Scalar quantity with no direction, such as 50km/h.

Average Speed

• (S) is distance travelled/change in time = (l)/Δt = (l)/(t2 – t1).
• The winner of a race has is the quickest average speed.

Interntaneous Speed

• Speed of an object at a given moment thatfluctuates above and below average speed.

Velocity

• Rate of motion in a specific direction.
• Rate of motion in a specific direction E.g., 50km.h North (SI units still m/s).
• Vector quantity represented by an arrow with the resultant vector or derived yby trigonometry.

Velocity: Average

• V= change in position over change in time = p2-p1/t2-t1.

Acceleration

• Rate of change of velocity.
• Rate of change of velocity, such as, 2m/s2.
• As time passes, the velocity increases 2m/s resulting in an object at 10m/s and increases to 12m/s in 1 sec.
• Measured as a vector quantity (magnitude and direction).

Acceleration: Vectors

• Vector is described by an arrow and resultant vector is determined by trig.
• Accelerating objects speed up or slow down, start, stop, or change direction.
• Deceleration is negative acceleration.
• The direction of motion is not necessarily the direction of acceleration.

Acceleration: Average

• a= change in velocity over change in time: (v2-v1)/(t2-t1)

Acceleration: Instantaneous

• Acceleration of an object at a given instant.

Acceleration: Uniform

• Acceleration of an object is constant due to a net external force and is represented by the motion of a projectile in a verticle direction: -9.81m/s^2.

Projectile Motion

• Occurs when an object is propelled in air/dropped and only has forces of gravity and drag (air resistance) acting on it (eg any ball in flight).
• Since drag is negligable (a=g= -9.81m/s^2), the motion os expressed by vaious equations.
• A ball's parabolic path is symmetrical where vertical velocity has the same value (oppisite sign), V2 = -V1 where peak height v=0.

Projectile Motion: Verticle

• Verticle positioning determined by by2 = y1 + v1∆t + 1/2 g(∆t)2.
• Determining velocity = v2 = v1 + g∆t AND v2 = v12 + 2g∆y.

Vertical Motion of Objects Dropped

• Start position (y1) and start velocity (v1) are = 0, so equations are simplified to: y2=y1 + VA + 1½ g(A1)² - y₂ = 1½ 8(A1)²; V₂ = V1 + 8At - V₂ = gAt; V₂ =V+2gAy — v² = 2gAy
• 2

Projectile Motion: Launched Objects

• The peak velocity is (v2) and is 0, this can be used to maximize height or time from the simplified velocity equations.

Vertical Motion Equations for Peak Values

• Time to peak and max height = v2=v1+gt →0 = v₁ + gAt; - Max height of time= V2 =V2+2gAy=0=v₁ + 2gAy.
• Overall Flight time = Flight time: V2 =V₁+gAt.

Horizontal Projection

• Horizontial velocityis constatnt v =V2=v1 = constant.
• Since velocity is constant, that means the horizontail acceleration is 0.
• The horizontial position is found via x2 = x1 + VA.

Newton's First Law: Inertia

• Law of inertia: An object stays in a state of rest, or constant velocity in a straight line, unless acted on by an external force (v = constant IF ΣF = 0 and vice versa).
• Inertia can be applied to the resultant motion but also to motion components (ΣFx and ΣFy ).Interpretation.
• An objects at rest and ΣF = 0, will remain at rest; objects in motion with ΣF = 0 continue at a constant speed. A stationary and non-accelerating object ∑F= 0 (basis for static equilibrium and for the principle of momentum conseravation).

Linear Momentum

• (L) is the product of an object's mass (m) and instantaneous linear velocity (v L=mv).
• L and v proportionally in a directionally proportionate way, vector quanties: SI unite are kgms.
• The greater an object's mass or speed the greater the momentum.
• If mass is constant, the linear momentum is proportional to to linear momentum (if EF= 0 both velocity and momento are constantly proportionate).

The Conservation of of Momento

• The definition is at total momentum of a system is constant if ∑F=0. Linitial = Final is a colision equation: my + mv = m + mzvf.
• Linear monemtum describes the collisions of objects.

Types of Collisions

• Elasticity is colision detemrined by restitution (e) that can be defined as rati of relative velocity B= (V₁-V₂end/ (V₁-V₁initial, Objects 1 and 2 are represented by v₁ and v₂.

Restitution: The Coefficient

• Reported as an absolute value with no referenc eto + or-.
• Ranges vaules from 0 to 1( perfectly elastic collision).
• Coefficients lack units and are also used to in ball spoorts measure bounciness ( regulated in leagues ). The ball is dropped onto an object's and the restitution is calculated via e=\sqrt{\frac{bounce height}{drop height}}.
• Perfect elastic colection = \sqrt{\frac{bounce height}{drop height}} : object colide and bouce.

Acceleration: The Law of Motion (Newtons 2nd)

• The effect of applied force: the change of motion is directly proportional to the (net external) force impressed in the object, made in direction of the force (ΣF=ma where; ΣF = net external force, m=ass of the object, a-instantaneous acceleration).
• An equal and opposite force back is applied on the object proportional its mass (ΣF=ma.
• The component vectors: are (ΣFx= m ax ) are force causes acceleration || or is the effect of forcesΣFy = may.
• Newtons 1º law is a special case of the 2nd law where ΣF = 0.

Acceleration: Verticle Calculation

• ∑Fy and a + ΣFy=(ground reaction force)+(the object weight with a negative), ΣFy = R+ (-W),R+ (-W)=(m, a), then, ΣFy = R + -W+/ and R+ (-W)+/= ay.

Acceleration: Horizontal Acceleration

• ΣF=max
• Where the only effects are a push or pull (ΣF=ground reaction force)+(the object weight with a negative)ΣFx=Px+(+Ef),Px+(-ef)=m, af (if other forces +_X is added).

The Second Low of Motion(Newtons)(Motion Sides)

• To move up you need above the gravitational force to accelrate up (W).
• To move you would exceed forces friction of sides by an mount (Ef).
• To acceleration and object forces must be greater a(Fx is always smaller than ) R (W).R is equal, to (W the the is a (u) that its constant).

The Second Low of Motion(Impulse and Momentum )

• N2L provides the N external. force for the external of the (a)at point a specific pont in time.
• In sport there focus with external the final results during for a time.
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• Because a force makes a constant in speed, and a force

Accelerates and Decrease the Velocity (Decreases and increase Momentum)

• Increase: ∑F Δt = m (V₂ - V₁) ↑ (increases moment) -
• Decrease: Two possible the for formula; ∑FΔt = m (V₂ - V₁) (reduces velocity) ∑FΔt = m (V₂ - V₁) ( minimizes impulse by maximizing time for reduction).

Third Law of Action and Reaction (Newtons)

• For each Action the is a equal oppising is reaction; this is how forces in existing an pairs are
• And when these pairs occur, the affects of time vary due to and forces Frictional.

(U, ENERGIES, AND FORCES ) Works

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• force U, the are it is need is amount know the object.
• work Isometric from static and and is also and and from but different this work is a.

( KE, FORCES, AND ENERGIES ) Kinetic

• Kinetic energy and that it half is and and that there force
• Mech kinetic PE PE U= force (U).
• in and from gravity from but all that the.