Engineering Physics 1: Mechanics - Units & Measurements

Introduction to Engineering Physics 1: Mechanics

• This set of notes is based on "Physics for Scientists and Engineers" by Serway and Jewettt, 9th edition.
• The notes cover Mechanics, Waves, vibrations, Electricity, magnetism, Thermodynamics, light and optics, and Modern Physics.
• The notes only contain essential course information, and the textbook should be consulted for a better understanding.
• The textbook is the source of all examples and practice problems.
• Lectures and lab work provide skills in analysing physical problems, developing solutions, conducting experiments, and drawing conclusions.
• A lab manual will be given to complement lectures.

Units and Measurements

• Physics relies on experimental measurements to determine basic laws, which are then refined through further experimentation.
• Measurements relate to physical quantities like length, mass, and velocity.
• Laws of nature are expressed mathematically using these quantities.

Fundamental Quantities in Mechanics

• Length, mass, and time are fundamental quantities.
• All other quantities can be expressed in terms of these.

SI System of Units

• The SI system is used for specifying physical quantities with standard units.
• Metre (m) is the unit for length.
• Kilogram (kg) is the unit for mass.
• Second (s) is the unit for time.
• Kelvin (K) is the unit for temperature.
• Ampere (A) is the unit for electric current.

Length Measurements

• Length measures the distance between two points in space.
• Common metric units include millimetres (mm), centimetres (cm), and metres (m), with the metre being the SI unit.
• Lab measurements are typically in mm or cm but should be converted to metres for reports:
• 1 m equals 1000 mm.
• 1 m equals 100 cm.
• 1 mm equals 0.001 m.
• 1 cm equals 0.01 m.

Mass Units

• The SI unit for mass is the kilogram (kg).
• Gram (g) and milligram (mg) are often used initially in the laboratory.
• Mass conversions are:
• 1 kg equals 1000 g.
• 1 kg equals 1,000,000 mg.
• 1 g equals 0.001 kg (10^-3 kg).
• 1 mg equals 0.000001 kg (10^-6 kg).

Time Units

• The SI unit for time is the second (s).
• Important time conversions:
• 1 min equals 60 s.
• 1 hour equals 60 min or 3600 s.
• 1 ms equals 10^-3 s.
• 1 μs equals 10^-6 s.

Dimensional Analysis

• Dimension refers to the physical nature of a measurement, focusing on length (L), mass (M), and time (T) in mechanics.
• Use brackets [ ] to denote the dimension of a variable; for example, [x] = L for distance.
• Dimensions on both sides of an equation must match.
• Speeds are represented as [v] = L/T.
• Acceleration is represented as [a] = L/T².

Determining Relationships with Dimensional Analysis

• To find the relationship between variables, express each variable in terms of its dimensions (L, M, T).
• Compare the powers of each dimension on both sides of the equation to derive the relationship.

Significant Figures

• Maintain the correct number of digits after the decimal point in calculations.
• Rounding off to three digits after the decimal point is acceptable for engineering practice.
• Measurement accuracy is limited by the instrument used and results in significant figures.

Motion in One Dimension

• An exploration into objects moving in a straight line is called kinematics dealing with motion without consideration for what is causing the motion.
• Treat all objects as point-like particles for these analyses.

Position, Velocity, and Speed

• A reference frame or coordinate system is needed to specify the position of an object.
• Position is the distance of an object from the origin, and can be positive or negative.
• An object's position changes as it moves with time.
• Displacement (∆x) is the change in position between two points.
• Displacement is a vector with magnitude and a sign indicating direction.

Distance

• Total distance d is travelled between an initial time ti and final time tf.
• This is always positive and a scalar quantity.

Average Velocity and Average Speed

• Average velocity (vx,avg) is the change in position divided by the change in time.
• Average velocity is expressed as: vx,avg = (xf - xi)/(tf - ti) = ∆x/∆t and is a vector quantity.
• Average speed (vavg) is the total distance travelled divided by the time interval: vavg = d/∆t and is a scalar quantity.

Instantaneous Velocity and Speed

• Instantaneous velocity (vx) equals the limit of ∆x/∆t as ∆t approaches 0: vx = lim(∆t→0) ∆x/∆t.
• Velocity is the slope of the tangent to the position vs. time graph.
• Velocity is a vector quantity.
• Instantaneous speed is the magnitude of v.

Constant Velocity Motion

• Constant velocity means v is the same at all times.
• When velocity is constant:
• The position-time graph is a straight line.
• Instantaneous velocity equals average velocity.
• Position is given by: x = xi + vxt.

Average and Instantaneous Acceleration

• Average acceleration (ax,avg) is the change in velocity divided by the change in time.
• Expressed as: ax,avg = (vf - vi)/(tf - ti) = ∆vx/∆t.
• Is a vector quantity.

Constant Acceleration

• Expressed as: ax = lim(∆t→0) ∆vx/∆t.

Instantaneous Acceleration

• The slope or gradient of the tangent to the velocity vs. time graph indicates acceleration with time t.
• This is a vector quantity.

Constant Acceleration Motion

• Velocity as a function of time: vxf = vxi + axt.
• Average velocity: vx,avg = (vxf + vxi)/2.
• Position as a function of time:
• xf = xi + vx,avgt.
• xf = xi + (vxi + vxf)t /2 .
• xf = xi + vxit + axt2 /2.
• Velocity as a function of position: vx2f = vx2i + 2ax(xf - xi).
• For constant velocity (ax = 0):
• vxf = vxi = vx.
• xf = xi + vxt.

Vectors

• In two dimensions, position is specified using a two-axis coordinate system.

Coordinate System

• The Cartesian coordinate system consists of two perpendicular axes.
• Any point P within this coordinate system is represented by coordinates(x,y).
  • Distance (r) from the origin to point P: r = √(x² + y²).
  • Polar coordinates express P with distance r and angle θ:
• x = r cos θ.
• y = r sin θ.
• tan θ = y/x.

Scalar Quantities

• Specified by a single number and a unit.
• Examples: distance, temperature, mass and energy.

Vector Quantities

• Requires two numbers: magnitude and direction as well as a direction.
• Examples: displacement, velocity, acceleration, force, electric and magnetic fields.

Vector Notation

• An arrow is used over a letter to denote a vector, (example A’ ) and the magnitude is given by A:

Properties of Vectors

• Equality: Two vectors A’ and B’ are equal if they hold the same magnitude + direction. (A’ = B’)
• Addition: Vector addition with A’ + B’ gives resultant ©.

Vector Diagrams

• Vector diagrams are the most powerful tools for geometric solutions

Scalar Multiplication

• Given a vector A’ + scalar number (m) where m is the vector Am whose magnitude is that of scaled(m).

Component Vectors

• A vector can be expressed as the sum (+/-) of each respective co-ordinate.

Components of a Vector and Unit Vectors

• Vectors can be represented in terms of components, i.e. it's projection onto the i and y axis using a rectangular system
• Ex, A = Ax + Ay
• Ax - component of A + x axis
• ay - component of A + y axis

Magnitude

• Given by vector A + vector components
• Ax’ = Acose
• Ay = AsinO

Relationship between Coordinates system

• 0 = tan-1 (Ay/Ax) Magnitude A is always positive, components, can either be positive or negative depending on placement

Unit Vector

• Vector that denotes magnitude- unit vector that lies among the x axis and y axis (and z axis)

Symbols

• i = unit vector for x axis
• J = unit vector for y axis
• K = unit vector for z axis The magnitude is represented by modulus of a given function, hence absolute sign. The vector in 2D is written as A= A’xi + A’yj

Motion

• A- xi +Ayj

Special Vector

• r- vector that special position
• consider an object that has location that is fixed on a particular coordinate.
• r = xi+yj

Adding vectors

• Using its compoenents the total equation is Let A = Axi + Ayj, and B = Bxi + ByjR=A+B = (Axi + Ayj) + (Bxi + Byj) = (Ax + Bx) + (Ay + By)j -You simply add their compeonets- R’x = Ax + By
• Rx = Ax + Bx
• Ry = Ayby R= Rxi + Ry

Motion in Two Dimensions

• Two coordinates, x & y, are needed to measure the position of objects.
• Vector r is used to denote the position

Position, Velocity + Acceleration Vectors

• Position refers to x + y co-ordinate velocity
• Change of + in position in vector form (triangle)

Vector Rules:

• Average Velocity/Acceleration: the velocity is found by average velocity/function = ar and magnitude Tangential and radial acceleration

Forces

• (F) can cause change in direction, acceleration

Constant Acceleration, and 1D equations used to derive vector, time - Acceleration can be tangent to one another (the path, and speed at which its changed) or it cannot be

Newton's Law

• First introduced
• concept of force is examined Newton's law- these laws explain what initiates of objects motion when we observe certain patterns = forces First Law inertia, inertia fram second - F a 3rd action + reaction the above provide how we know that objects do move

Force friction

• Occurs when surfaces are touching each other and there're forces of opposing each other (roughness)
  • 2 types- kinetic and static

Equilibrium, free diagrams, equal force.

• When they are in equilibrium, they either net force or the component (is either at 0) no movement We can analyze forces and their respective force- friction can play a great role