Principles of Engineering

Projectile Motion

• Speed: Distance traveled per unit of time.
• Velocity: Object's speed and direction of motion.
• Acceleration: Change in velocity over a certain period of time.
• Gravity tends to pull all objects toward the center of the earth.
• Gravitational Acceleration: Constant describing the acceleration of any object falling toward the earth.
  • $-9.8 m/sec^2$
  • $-32 ft/sec^2$
• Projectile: Any moving object upon which the only active force is gravity.
• Gravity pulls all projectiles toward the center of the earth at the same rate.
• Firing Angle ($\theta$): Angle at which the projectile left the cannon, measured in degrees.
• Initial Velocity ($V_i$): Angular speed of a projectile at the start of its flight.
• Calculating Initial Velocity:
  • $V_i = \frac{-gx}{sin(2\theta)}$
    • Where:
      • $V_i$ = Initial Velocity
      • g = Gravitational Acceleration
      • x = Horizontal Distance Traveled
      • $\theta$ = Firing Angle
• Calculating Horizontal Distance:
  • $x = \frac{V_i^2 sin(2\theta)}{-g}$
    • Where:
      • $V_i$ = Initial Velocity
      • g = Gravitational Acceleration
      • x = Horizontal Distance Traveled
      • $\theta$ = Firing Angle
• Calculating Firing Angle:
  • $\theta = \frac{1}{2} sin^{-1} \frac{-gx}{V_i^2}$
    • Where:
      • $V_i$ = Initial Velocity
      • g = Gravitational Acceleration
      • x = Horizontal Distance Traveled
      • $\theta$ = Firing Angle

Statistics

• Statistics: The collection, evaluation, and interpretation of data.
• Descriptive Statistics: Describe collected data.
• Inferential Statistics: Generalize and evaluate a population based on sample data.
• Categorical or Qualitative Data: Values that possess names or labels (e.g., color of M&M's, breed of dog).
• Numerical or Quantitative Data: Values that represent a measurable quantity (e.g., population, number of M&M's).
• Data Collection Sampling:
  • Random
  • Systematic
  • Stratified
  • Cluster
  • Convenience
• Graphic Data Representation:
  • Histogram: Frequency distribution graph.
  • Frequency Polygons: Frequency distribution graph.
  • Bar Chart: Categorical data graph.
  • Pie Chart: Categorical data graph.
• Measures of Central Tendency:
  • Mean:
    • Arithmetic average.
    • Sum of all data values divided by the number of data values within the array.
    • Most frequently used measure of central tendency.
    • Strongly influenced by outliers—very large or very small values.
  • Median:
    • Data value that divides a data array into two equal groups.
    • Data values must be ordered from lowest to highest.
    • Useful in situations with skewed data and outliers (e.g., wealth management).
  • Mode:
    • Most frequently occurring response within a data array.
    • May not be typical.
    • May not exist at all.
    • Modal, bimodal, and multimodal.
• Data Variation:
  • Range: Difference between the lowest and highest data value.
  • Standard Deviation: Square root of the variance.
• Standard Deviation – Sample vs. Population:
  • Population Standard Deviation:
    • Represents a parameter, not a statistic.
    • Gives researchers an amount of dispersion of data for an entire population of survey respondents.
  • Sample Standard Deviation:
    • Estimates the standard deviation of a population based on a random sample.
    • A statistic that measures the dispersion of the data around the sample mean.
• Sample Standard Deviation Calculation:
  1. Calculate the mean.
  2. Subtract the mean from each value and then square it.
  3. Sum all squared differences.
  4. Divide the summation by the number of values in the array minus 1.
  5. Calculate the square root of the product.
• Population Standard Deviation Calculation:
  1. Calculate the mean.
  2. Subtract the mean from each data value and square each difference.
  3. Sum all squared differences.
  4. Divide the summation by the number of data values.
  5. Calculate the square root of the result.
• Graphing Frequency Distribution:
  • Numerical assignment of each outcome of a chance experiment.
  • The calculated likelihood that an outcome variable will occur within an experiment.

Probability

• Probability: The calculated likelihood that a given event will occur.
• Methods of Determining Probability:
  • Empirical: Experimental observation.
  • Theoretical: Uses known elements (e.g., coin toss, die rolling).
  • Subjective: Assumptions (e.g., I think that . . .).
• Probability Components:
  • Experiment: An activity with observable results.
  • Sample Space: A set of all possible outcomes.
  • Event: A subset of a sample space.
  • Outcome / Sample Point: The result of an experiment.
• Probability:
  • A way of communicating the belief that an event will occur.
  • Expressed as a number between 0 and 1 (fraction, percent, decimal, odds).
  • Total probability of all possible events totals 1.
• Relative Frequency:
  • The number of times an event will occur divided by the number of opportunities.
  • Expressed as a number between 0 and 1 (fraction, percent, decimal, odds).
  • Total frequency of all possible events totals 1.
  • $\text{Relative frequency of outcome x} = \frac{\text{Number of events with outcome x}}{\text{Total number of events}}$
• Binomial Process:
  • Each trial has only two possible outcomes (yes-no, on-off, right-wrong).
  • Trial outcomes are independent.
• Bernoulli Process:
  • P = Probability
  • x = Number of times for a specific outcome within n trials
  • n = Number of trials
  • p = Probability of success on a single trial
  • q = Probability of failure on a single trial
  • ! = factorial—product of all integers less than or equal
• Probability Distribution What is the probability of tossing a coin three times and it landing heads up two times?
• Law of Large Numbers:
  • The more trials that are conducted, the closer the results become to the theoretical probability.
• Independent events occurring simultaneously:
  • Product of individual probabilities.
  • If events A and B are independent, then the probability of A and B occurring is:
    • $P(A \text{ and } B) = P(A) \cdot P(B)$
• Independent events occurring individually:
  • Sum of individual probabilities.
  • If events A and B are mutually exclusive, then the probability of A or B occurring is:
    • $P(A \text{ or } B) = P(A) + P(B)$
• Independent event not occurring:
  • 1 minus the probability of occurrence
  • $P = 1 - P(A)$
• Conditional Probability:
  • $P(E|A)$ = Probability of event E, given A
• Conditional Probability:
  • Probability of two events A and B both occurring =
    • $P(A \text{ and } B) = P(A|B) \cdot P(B) = P(B|A) \cdot P(A)$
    • If A and B are independent, then $P(A \text{ and } B) = P(A) \cdot P(B)$
• Bayes’ Theorem:
  • Calculates a conditional probability, based on all the ways the condition might have occurred.
  • $P( A | E )$ = probability of A, given we already know the condition E
• Expected Value:
  • $\langle x \rangle = \Sigma P<em>i x</em>i$
  • $\langle x \rangle$ = expected value
  • $x_i$ = the value of an element in a list
  • $P_i$ = Probability of element i

Hydraulic Power

• Hydraulic Power: The use of a liquid flowing under pressure to transmit power from one location to another.
  • Liquid in a hydraulic system behaves like a solid since it compresses very little.
• Benefits of Hydraulic Systems:
  • Use a relatively incompressible liquid
  • Have a slower, smoother motion
  • Are generally more precise
  • Lubricate naturally
  • Are not as clean as pneumatics when leakage occurs
  • Often operate at pressures of 500–5000 psi
  • Generally produce more power
• Early Hydraulic Uses:
  • Water Wheels: Create rotational motion.
  • Roman Aqueducts: Delivered water to buildings, agricultural fields, and fountains.
• Hydrodynamic Systems:
  • Fluid is in motion
  • Force and energy are transmitted by flow
• Hydrostatic Systems:
  • Fluid does not flow quickly or continuously
  • Fluid is pressurized
  • Force and energy transmitted by pressure.
  • Most common in industrial settings.
  • Pascal’s Law: Pressure exerted by a confined fluid acts undiminished equally in all directions.
• Liquid Flow:
  • Flow Rate: The volume of fluid that moves through a system in a given period of time.
  • Flow Velocity: The distance the fluid travels through a system in a given period of time.
  • $Q = A \cdot v$
    • Where:
      • Q = Flow Rate
      • v = Flow Velocity
      • A = Area
• Mechanical Advantage:
  • $MA = \frac{F<em>{\text{out}}}{F</em>{\text{in}}}$
    • Where:
      • $F_{\text{out}}$ = Force at the output
      • $F_{\text{in}}$ = Force at the input
• Bernoulli’s Principle: An increase in velocity results in a decrease in pressure. Likewise, a decrease in velocity results in an increase in pressure.
• Viscosity:
  • The measure of a fluid’s thickness or resistance to flow.
  • Crucial for lubricating a system.
  • Decreases as temperature increases.
• Common Hydraulic System Components:
  • Reservoir
  • Pump
  • Cylinder
  • Valve
  • Transmission Lines
  • Directional Control Valve
  • Filter

Pneumatic Power

• Pneumatics: The use of a gas flowing under pressure to transmit power from one location to another.

  • Gas in a pneumatic system behaves like a spring since it is compressible.

• Pneumatic Systems:

  • Use a compressible gas
  • Possess a quicker, jumpier motion
  • Are not as precise
  • Require a lubricant
  • Are generally cleaner
  • Often operate at pressures around 100 psi
  • Generally produce less power

• Early Pneumatic Uses:

  • Bellows: Tool used by blacksmiths and smelters for working iron and other metals.
  • Otto von Guericke: Showed that a vacuum can be created.
  • America’s First Subway: Designed by Alfred Beach.

• Properties of Gases:

  • Gases are affected by 3 variables:
    • Temperature (T)
    • Pressure (p)
    • Volume (V)
  • Gases have no definite volume
  • Gases are highly compressible
  • Gases are lighter than liquids

• Absolute Pressure:

  • Gauge Pressure + Atmospheric Pressure = Absolute Pressure

• Absolute Temperature:

  • $\text{Absolute Zero} = -460 \text{°F}$
  • $\text{Absolute Temperature is measured in degrees Rankine (°R)}$
  • $\text{°R} = \text{°F} + 460$

• Pascal’s Law:

  • Pressure exerted by a confined fluid acts undiminished equally in all directions.

  • Pressure: The force per unit area exerted by a fluid against a surface

    • $P = \frac{F}{A}$
      • Where:
        • P = Pressure
        • F = Force
        • A = Area

• Perfect Gas Laws:

  • The perfect gas laws describe the behavior of pneumatic systems
    • Boyle’s Law
    • Charles’ Law
    • Gay-Lussac’s Law

• Boyle’s Law:

  • The volume of a gas at constant temperature varies inversely with the pressure exerted on it.
    • $p<em>1 (V</em>1) = p<em>2 (V</em>2)$

• Charles’ Law:

  • Volume of gas increases or decreases as the temperature increases or decreases, provided the amount of gas and pressure remain constant.
    • $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$
    • Note: T1 and T2 refer to absolute temperature.

• Gay-Lussac’s Law:

  • Absolute pressure of a gas increases or decreases as the temperature increases or decreases, provided the amount of gas and the volume remain constant.
    • $\frac{p<em>1}{T</em>1} = \frac{p<em>2}{T</em>2}$
    • Note: T1 and T2 refer to absolute temperature. p1 and p2 refer to absolute pressure.

• Common Pneumatic System Components:

  • Receiver Tank
  • Compressor
  • Transmission Lines
  • Cylinder
  • Pressure Relief Valve
  • Directional Control Valve
  • Filter
  • Regulator
  • Drain

Fluid Power Introduction

• Fluid Power: The use of a fluid to transmit power from one location to another
• Hydraulics: The use of a liquid flowing under pressure to transmit power from one location to another
• Pneumatics: The use of a gas flowing under pressure to transmit power from one location to another
• Why Use Fluid Power?
  • Multiplication and variation of force
  • Easy, accurate control
  • One power source controls many operations
  • High power / low weight ratio
  • Low-speed torque
  • Constant force and torque
  • Safe in hazardous environments
• Basic Fluid Power Components
  • Reservoir / Receiver – Stores fluid
  • Fluid Conductors – Pipe, tube, or hose that allows for flow between components
  • Pump / Compressor – Converts mechanical power to fluid power
  • Valve – Controls direction and amount of flow
  • Actuators – Converts fluid power to mechanical power
• Fluid Power Physics
  • Energy The ability to do work
    • Energy Transfer From prime mover, or input source, to an actuator, or output device
• Fluid Power Physics
  • Work
    • Force multiplied by distance
    • Measured in foot-pounds (ft-lb)
• Fluid Power Physics:
  • Power
    • The rate of doing work
    • Work over time in seconds
• Fluid Power Principles:
  • Horsepower
    • Hydraulic power is given by:
      • Power = flow x pressure drop,
    • Horsepower is a common unit for power
      • $1 hp = 1714 \frac{gal}{min} \cdot 1 psi$
• Fluid Power Principles:
  • Heat:
    • Law of conservation of energy states that energy can neither be created nor destroyed, although it can change forms.
    • Energy not transferred to work takes the form of heat energy.
• Fluid Power Principles:
  • Torque:
    • Twisting force
      • force x distance
    • Measured in foot-pounds
• Fluid Power Principles:
  • Torque:
    • The twisting force applied by a hydraulic or pneumatic motor
    • Motor rpm at a given torque specifies power usage or horsepower requirement
• Fluid Power Principles:
  • Flow:
    • Makes actuator operation possible
    • To extend the cylinder, flow must be directed into port B.
    • To retract the cylinder, flow must be directed into what port?
    • The cylinder retracts when flow is directed into Port A.
• Fluid Power Principles:
  • Rate of Flow:
    • Determines actuator speed
    • Measured in gallons per minute (gpm)
    • Generated by a pump
• Fluid Power Principles:
  • With a Given Flow Rate
    • Actuator volume displacement directly affects actuator speed
    • The less volume to displace, the faster the actuator
• Fluid Power Principles:
  • Pressure:
    • Overcomes the resistance to flow
    • Pumps produce flow by pressurizing the fluid
      • A pump can create greater pressure at lower flow rate, so if you restrict the flow from the pump, greater pressure will result.
    • All points of resistance in series within a system contribute to total system resistance, including long runs of pipe, elbows, etc.
• Fluid Power Principles:
  • Definition of pressure
    • Relationship between force, pressure, and area
    • Blaise Pascal developed concepts about pressure in the 1640’s.
    • The SI unit for pressure is the pascal (Pa).
      • $1 Pa = 1 \frac{N}{m^2}$
• Fluid Power Principles:
  • Pascal’s Law
    • Pressure applied on a confined fluid at rest is transmitted undiminished in all directions and acts with equal force on equal areas and at right angles to them.
• Fluid Power Schematics:
  • Schematics
    • Line drawing made up of a series of symbols and connections that represent the actual components in a hydraulic system
• Fluid Power Schematics:
  • Symbols
    • Critical for technical communication
    • Not language-dependent
    • Emphasize function and methods of operation
• Fluid Power Schematics:
  • Lines
    • Continuous lines indicate working, pilot supply, return or electrical lines
    • Dashed lines indicate a pilot, drain, purge, or bleed line
    • Flexible lines indicate a hose which usually connects moving parts
    • Crossing lines use loops at cross over
    • Lines joining may use a dot at the junction
    • Components (like this filter) inserted into lines
• Fluid Power Schematics:
  • Reservoirs
    • Vented reservoirs are shown as rectangles without top lines
    • Pressurized reservoirs are shown as capsules
• Fluid Power Schematics:
  • Pumps
    • Rotary devices are shown as circles
    • Pumps having a triangle indicating where the energy is leaving component
• Fluid Power Schematics:
  • Flow Control Valves
    • An upper and lower arc symbolize a fixed orifice flow control valve.
    • An arrow through the arcs indicate an adjustable orifice
• Fluid Power Schematics:
  • Directional Control Valves
    • A three-position valve is shown with 3 envelopes
    • Arrows indicate possible flow direction through valve
• Fluid Power Schematics:
  • Check Valves
    • Check valves are drawn with small circles inside an open triangle
    • Free flow is opposite the direction the triangle is pointed
• Fluid Power Schematics:
  • Motors
    • Energy triangle points into the circle indicating fluid energy entering
    • Two energy triangles indicate a bi-directional or reversible motor
• Fluid Power Schematics:
  • Cylinders
    • Single Acting (one line)
    • Double acting (two lines)

Properties of Materials

• Material Testing
  • Reproducible evaluation of material properties
    • Material response to varying loading conditions, including magnitude, cycling, and mode
      • Dynamic Testing
        • Material response to constant loading
      • Static Testing
• Static Material Testing
  • Strength
  • Deformation
  • Fracture
  • Design requirement compliance
    • Tensile test
    • Compression test
    • Hardness test
  • Evaluation of Material
    • Standardized Tests
• Tensile Test
  • Uniaxial
    • A straight line axial force is applied to a test sample (typically in the y-axis).
  • Destructive
    • Force is applied until sample fails
• Tensile Test
  • Standard Test Sample (dog bone)
    • Ensures meaningful and reproducible results
    • Uniform cross section
• Tensile Test Procedure
  • Dog bone is created to test specifications
  • Dog bone is secured in tester
• Tensile Test Procedure
  • A tension force (F) is applied to the dog bone until failure occurs.
  • Simultaneously the applied tension force (F) and dog bone elongation (δ) are recorded.
  • A plot is created from the stored load elongation data.
• Tensile Test Data
  • Load-elongation results are dependent upon sample size.
• Tensile Test Data
  • To eliminate test results based on sample size, calculate sample stress.
    • Divide load (F) by the original test sample cross-sectional area (A0).
    • Stress is load per unit area.
• Tensile Test Data
  • Manipulating Elongation Results
    • To eliminate test results based on sample size, calculate sample strain.
      • Strain ($\epsilon$) - the amount of stretch per unit length
      • Elongation ($\delta$) under load divided by the original Length (L0)
      • $\epsilon = \frac{\delta}{L_0}$
• Tensile Test—Stress-Strain Curve
  • Initial response is linear.
    • Stress and strain are proportional to one another.
      • Elastic Range
      • Proportional Limit (The stress at which proportionality ceases)
• Tensile Test—Stress-Strain Curve
  • Modulus of Elasticity (E)
    • The proportional constant (ratio of stress and strain)
    • A measure of stiffness—The ability of a material to resist stretching when loaded
    • An inherent property of a given material
    • If the load is removed, the test sample will return to its original length.
      • The response is elastic or recoverable
• Tensile Test—Stress-Strain Curve
  • Elastic Limit
    • Uppermost stress of elastic behavior
    • Elastic and proportional limit are almost identical, with the elastic limit being slightly higher.
• Tensile Test—Stress-Strain Curve
  • Resilience
    • The amount of energy per unit volume that a material can absorb while in the elastic range
    • Area under the stress-strain curve
• Tensile Test—Stress-Strain Curve
  • Yield Point
    • When the elastic limit is exceeded
    • A very small increase in stress produces a much greater strain.
    • Most materials do not have a well-defined yield point.
• Tensile Test—Stress-Strain Curve
  • Offset Yield Strength
    • Defines the stress required to produce a tolerable amount of permanent strain
    • Common value is 0.2%
• Tensile Test—Stress-Strain Curve
  • Plastic Deformation
    • Unrecoverable elongation beyond the elastic limit
    • When the load is removed, only the elastic deformation will be recovered.
• Tensile Test—Strength Properties
  • Plastic deformation represents failure.
  • Part dimensions will now be outside of allowable tolerances.
• Tensile Test—Stress-Strain Curve
  • Test sample elongation
  • Cross-sectional area decreases
  • Load bearing ability increases
    • The material is getting stronger
• Tensile Test—Stress-Strain Curve
  • New weakest point is stretched and becomes stronger, and so on
  • This keeps occurring until the decrease in area overcomes the increase in strength
• Tensile Test—Stress-Strain Curve
  • Tensile Strength
    • Load bearing ability peaks
    • Force required to continue straining the test sample decreases
  • Weakest location at the peak continues to decrease in area—Necking
• Tensile Test—Stress-Strain Curve
  • Failure
    • If continued force is applied, necking will continue until fracture occurs
  • Ductility
    • Amount of plasticity before fracture.
    • The greater the ductility, the more a material can be deformed.
• Tensile Test—Stress-Strain Curve
  • Brittleness
    • Material failure with little or no ductility
    • Lack of ductility, not lack of strength
• Tensile Test—Stress-Strain Curve
  • Toughness
    • Work per unit volume required to fracture a material
    • Total area under the stress-strain curve from test initiation to fracture (both strength and ductility)
• Compression Test
  • Stress and strain relationships are similar to tension tests—elastic and plastic behavior
  • Test samples must have large cross-sectional area to resist bending and buckling
  • Material strengthens by stretching laterally and increasing its cross-sectional area
• Hardness Testing
  • Resistance to permanent deformation
  • Resistance to scratching, wear, cutting or drilling, and elastic rebound
• Hardness Testing
  • Brinell Hardness Test
    • A tungsten carbide ball is held with a 500 lb force for 15 sec into the material.
    • The resulting crater is measured and compared
• Hardness Testing
  • Rockwell Test
    • A small diamond-tipped cone is forced into the test sample by a predetermined load.
    • Depth of penetration is measured and compared.

Forces in Trusses

• Forces:
  • Compression: A body being squeezed.
  • Tension: A body being stretched.
• Truss
  • A truss is composed of slender members joined together at their end points.
• Simple Truss
  • A simple truss is composed of triangles, which will retain their shape even when removed from supports.
• Pinned and Roller Supports
  • A pinned support can support a structure in two dimensions.
  • A roller support can support a structure in only one dimension.
• Solving Truss Forces
  • Assumptions:
    • All members are perfectly straight.
    • All loads are applied at the joints.
    • All joints are pinned and frictionless.
    • Each member has no weight.
    • Members can only experience tension or compression forces.
• Static Determinacy
  • A statically determinate structure is one that can be mathematically solved.
  • J = Number of Joints
  • M = Number of Members
  • R = Number of Reactions
  • $2J = M + R$
• Statically Determinate
  • A truss is considered statically determinate when the static equilibrium equations can be used to find the reactions on that structure.
• Equilibrium Equations
  • $\Sigma M = 0$
    • The sum of the moments about a given point is zero.
  • $\Sigma F_x = 0$
    • The sum of the forces in the x-direction is zero.
  • $\Sigma F_y = 0$
    • The sum of the forces in the y-direction is zero.
• Using Moments to Find RCY
  • A force that causes a clockwise moment is a negative moment.
  • A force that causes a counterclockwise moment is positive moment.
• Method of Joints
  • Use cosine and sine to determine x and y vector components.
  • Assume all members to be in tension.
• Method of Joints
  • $\Sigma F_Y = 0$
• Method of Joints
  • $\Sigma F_x = 0$

Moments

• Moment: The moment of a force is a measure of the tendency of the force to rotate the body upon which it acts.
• Terminology: The distance must be perpendicular to the force.
• Moments Formula:
  • $MO = d \cdot F$
• Units for Moments:
  • English Customary: Pound force (lbf) x Foot (ft) = lbf-ft
  • SI: Newton (N) x Meter (m) = N-m
• Rotation Direction:
  • In order to add moments, it is important to know if the direction is clockwise (CW) or counterclockwise (CCW).
    • CCW is positive
    • CW is negative
• Right-Hand Rule:
  • Curl your fingers to match the direction of rotation.
  • Thumb is pointing . . .
    • Up = Positive
    • Down = Negative
    • Toward You = Positive
    • Away from You = Negative
• What is Equilibrium?
  • The state of a body or physical system with an unchanging rotational motion.
    • Two cases for that condition:
  1. Object is not rotating OR
  2. Object is spinning at a constant speed
     • In either case rotation forces are balanced: The sum of all moments about any point or axis is zero.
     • $\Sigma M = 0$
       • $M<em>1 + M</em>2 + M_3 . . . = 0$

Force Vectors

• Vectors:

  • Have both a magnitude and direction
    • Examples: Position, force, moment
  • Vector Quantities

• Illustrating Vectors:

  • Vectors are represented by arrows
    • Include magnitude, direction, and sense

• Sense:

  • Indicated by the direction of the tip of the arrow

• Trigonometry Review:

  • Right Triangle
    • A triangle with a 90° angle
    • Sum of all interior angles = 180°
    • Pythagorean Theorem:
      • $c^2 = a^2 + b^2$

• Trigonometry Review:

  • $sin \theta = \frac{opp}{hyp}$
  • $cos \theta = \frac{adj}{hyp}$
  • $tan \theta = \frac{opp}{adj}$

• Trigonometry Application:

  • The hypotenuse is the Magnitude of the Force, F
  • In the figure here, The adjacent side is the x-component, Fx
  • The opposite side is the y-component, Fy

• Trigonometry Application:

  • $sin \theta = \frac{F_y}{F}$

  • $cos \theta = \frac{F_x}{F}$

  • $tan \theta = \frac{F<em>y}{F</em>x}$

  • $F_y = F sin \theta$

  • $F_x = F cos \theta$

• Resultant Force:

  • Draw the resultant force (FR)