Mechanics: Scalars and Vectors

• Mechanics is the physics branch focusing on object motion and the forces altering it
• Mechanics has two main branches: Dynamics and Kinematics

Kinematics

• Study of object motion, with no regard to forces

Dynamics

• Considers forces and their impact on object motion

Scalars

• Physical quantities described completely by just their magnitude
• Magnitude includes a number and a relevant unit
• Mass is a scalar quantity
• Distance, length, time, speed, energy, and temperature are examples of scalar quantities
• Scalar quantities can be added algebraically like numbers
• Example: 5 meters + 3 meters = 8 meters

Vectors

• Physical quantity requiring both magnitude and direction for complete description
• Displacement, velocity, acceleration, weight, and force are examples of vector quantities
• Vector example: A car moving at 90 km/h (25 m/s) towards north
• Velocity's magnitude is 25 m/s, and its direction is north
• Vectors need specific methods for addition, considering their directions

Representing Vectors

• Boldface letters represent vectors in textbooks (e.g., A, v)
• Vectors can be represented with an arrow above the letter due to writing limitations, for example A, v, F
• Magnitude of a vector is shown by an italic letter (without the arrowhead)
• Vectors are graphically represented by a straight line with an arrowhead at one end
• Line length is proportional to the vector's magnitude based on a chosen scale
• Arrow direction indicates vector's direction
• Two mutually perpendicular lines make up a way to represent direction
• North-south and east-west directions are represented by one line each
• Vector direction given relative to these lines
• Axes are also referred to as x-axis (horizontal) and y-axis (vertical)
• The point where the axes converge is the origin.
• Vectors are drawn with the origin as starting point in the given direction
• Direction relative to the x-axis represents an angle θ (theta)
• Angle is commonly gauged counter-clockwise from right side of x-axis

Vector Addition

• Two or more vectors can be combined into a single resultant vector
• Resultant vector has the same effect as combined vectors
• One method is the graphical approach

Head-to-Tail Rule

• Vectors are added by joining them such that one vector's head coincides with the tail of the next
• Resultant vector points from the tail of the initial vector to the head of the final vector

Rest

• A body is at rest if its position does not change with respect to its surroundings

Motion

• A body is in motion if it continuously changes its position relative to its surroundings
• Rest and motion are relative
• Example: A person in a moving train is at rest compared to other passengers but in motion relative to an outside observer

Types of Motion

• Three primary types of motion: translatory, rotatory, and vibratory

Translatory Motion

• When every particle of a body moves uniformly in the same direction
• Examples: a train or a car moving
• Can be further classified into linear, random, and circular motion

Linear Motion

• Movement along a straight line
• Example: A freely falling body

Random Motion

• Movement along an irregular path
• Example: The motion of a bee

Circular Motion

• Movement of a body along a circular path
• Example: A ball whirled at the end of a string or a Ferris wheel

Rotatory Motion

• When each point of a body revolves around a fixed axis
• Examples: An electric fan or a washing machine drum

Vibratory Motion

• When a body moves to and fro about a fixed position
• Example: A swing in a children's park

Distance

• Term referring to the length of the actual path of motion

Displacement

• Vector quantity representing the shortest distance between initial and final positions
• Displacement direction is from the initial to the final position

Speed

• Refers to how fast a body is moving
• Equals the distance covered in unit time
• Formula: Speed (v) = Distance (S) / Time (t)

Scalar Quantity

• Speed is a scalar quantity, with SI units of m/s or km/h

Instantaneous Speed

• Speed of a vehicle shown by the speedometer at any given instant

Average Speed

• Takes measurement from total distance divided by total time taken
• Formula: Average speed = Total distance covered / Total time taken

Velocity

• Velocity measures the direction of an object's motion
• The displacement (d) between initial and final positions in unit time = net displacement
• Formula: Average velocity (v) = Displacement (d) / Time (t)

Vector Quantity

• Velocity is a vector quantity, with SI units of m/s or km/h
• Velocity direction is the same as displacement direction

Uniform Velocity

• Implies constant speed and direction of a moving body
• Variable velocity, or non-uniform velocity happens when speed or direction fluctuates

Acceleration

• When the velocity of an object is increasing
• Examples include increase of speed when a car overtakes another one
• Opposite is declaration when velocity decreases

Vector Quantity

• Similar to velocity, acceleration is a vector quantity

Time Rate of Change of Velocity

• Acceleration is defined as the time rate of change of velocity
• Change in direction and magniture in velocity is how it occurs
• Acceleration is positive if velocity is increasing, negative if decreasing (deceleration or retardation)
• Average acceleration = (Change in velocity) / (Time taken)

Uniform Acceleration

• Achieved if the time rate of change of velocity is constant
• Variable or non-uniform acceleration happens when magnitude or direction fluctuates

Free Fall Acceleration

• Acceleration of a body falling freely influenced by Earth's gravity, given by 'g'
• Value of g is roughly 9.8 m/s², often approximated as 10 m/s² for simplicity
• Equations for freely falling bodies:
• v₁ = v + gt
• S = vt + (1/2)gt²
• 2gS = v² – v²

Free Fall Assumptions

• Motion is in a straight vertical line
• Only magnitudes of vector quantities are considered
• Acceleration is uniform
• Downward quantities as positive while any upward directional components are treated as negative

Force Concept

• Force is a push or pull
• Force can initiate, halt, or alter the magnitude or direction of a body's velocity

Contact Forces

• Force exerted by one another when contact is met
• Exerted forces like pushing, pulling, and twisting

Friction

• Resistance to motion, as one surface interacts with another resulting in contact

Drag

• Resistance force, acting against motion through a fluid, fluid can be liquid or air

Thrust

• Upward force exerted by a liquid on an immersed object

Normal Force

• Reaction force exerted by a surface on an object resting on it
• acts outward, pushing perpendicularly from the surface

Tension Force

• Experiences force from a rope when it's pulled

Elastic Force

• Returns materials to their initial shape after deformation

Non-Contact Forces

• Forces capable of acting despite physical seperation
• Sometimes called "action-at-a-distance" forces
• Always associated to a field

Gravitational Force

• Force examples: an apple falling from a tree
• This is based on an attractive force among objects with mass
• It can be calculated using Newton's gravitation
• Formula used is: F = G(m₁m₂) / r² which describes the long-range impact

Electrostatic Force

• Acts between two charged objects, where opposite charges attract and like charges repel
• It operates across long ranges

Magnetic Force

• Force exerted on magnetic materials like iron, cobalt, and nickel
• Can be visualized when iron pins attract without making physical contact

Strong and Weak Nuclear Forces

• Both nuclear forces act between subatomic particles

Fundamental Forces

• Gravitational force
• Electromagnetic force
• Strong nuclear force
• Weak nuclear force

Gravitational Force

• Weakest force, acting at long ranges

Electromagnetic Force

• Generates interaction among electrically charged particles
• Has long range effects

Strong Nuclear Force

• Retains atomic nuclei by joining neutrons and protons together
• Short range force

Weak Nuclear Force

• Drives disintegration of nucleus like beta decay

Free-Body Diagrams

• Show magnitudes and directions of forces acting on an object

Representation of Force

• Object illustrated using lines, with arrow outward from the center
• The direction that forces are acting upon
• Length of arrow represents magnitude of the force exerted

Newton's First Law of Motion

• Also termed Law of Inertia, it states: A body will maintain its inertia or uniform state, unless acted on by external forces
• All bodies moving on Earth tend to stop
• Provided there is contact, a force is exerted which will cause the objects to stop

Inertia

• Reflects the property of a body to maintain static state or uniform motion
• Mass of object measures its inertia

Newton’s Second Law of Motion

• It deals with applied force and acceleration
• States an body accelerates when force acts on it.
• Magnitude of accelaration means its is proportional to force, and inversely proportional to mass
• Formula used states: F = ma, with the SI unit for force is called newton (N)
• 1 N = 1 kg m s⁻²

Newton’s Third Law of Motion

• Interacting two bodies, one exerts force onto the other
• Reaction from other body is equal
• Force is exerted whenever there is an interaction between two bodies
• One presses a spring, there is exerted force from hand onto spring to act, in addition to spring’s feedback too

Limitations of Newton’s Laws of Motion

• Used for objects moving with high accuracy, in everyday situations
• For particles with more than light velocities, use work of Einstein

Mass

• Amount of mass
• Scalar meaning it will remain same, everywhere
• SI unit is kilogram (kg)

Weight

• Gravitational action force from object
• Variable
• Units are in Newtons (N)