# Statics/Print version

Statics

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# Introduction

## This Wikibook

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## Introduction

Statics is the branch of mechanics concerned with the study of forces and the effect of forces on a non-deformable, or rigid, system when the system is in a state of equilibrium.

This course is a crucial prerequisite for later areas, such as Dynamics and Properties of Materials. It utilizes principles of physics and calculus. It is fundamental in many different branches of engineering, from mechanical to civil engineering, and the principles of equilibrium, moment of inertia, and center of gravity will be revisited in more advanced fields. It is because an understanding of these topics is so crucial that statics does not cover a wide range of topics. Every problem will deal with some combination of two equations: the net forces being equal to zero, and/or net moments being equal to zero.

## Who is This Book For?

This book is for undergraduate students pursuing, or thinking about pursuing, a degree in engineering. As specified early, statics is needed in almost every field of engineering. It is not an introductory course, however. Statics does not attempt to reinvent the wheel, and is built firmly on the foundation of Physics. Newton's Laws are essential to statics. Mathematically, one should have a firm grip on vector algebra, though it is possible to solve many simple problems without the use of vectors. While it would benefit a student to understand calculus concepts, like the ability to calculate an area or a volume, most statics problems will involve simple geometries.

## How is this book organized

This book is organized to introduce concepts that will be later used in Dynamics and Properties of Materials. As such, it will begin by reviewing over vector mathematics and then introducing simple concepts, like Newton's Second Law, and then increasing the ways with which those concepts work together. Mathematically, this will involve increasing the number of forces, increasing the number of dimensions, and then expanding particle equilibrium to Rigid Body Equilibrium and Structural Equilibrium.

Finally, we will go over Moments of Inertia and Center of Mass. These concepts will be critical in higher level disciplines and a firm understanding will allow you to analyze more complex problems when the concepts of stress, strain, and acceleration are introduced.

# Measurement and Units

In statics and mechanics, units can be expressed in terms of three basic dimensions: length, mass, and time. All other units are created from combinations of these three basic units.

Force can be considered a fourth basic unit. Known as a derived measurement, it comes from Newton's 2nd Law:

${\displaystyle \ \ \!\mathrm {F} =\mathrm {m} \ \mathrm {a} \,\!}$

Here, force is defined as the amount of mass multiplied by the acceleration (length per second squared) that the mass achieves.

## International System of Units (SI Units)

In the SI system of units, the three specified base units are the units of length, mass and time. A fourth unit, that of force, is derived from the base units.

• The unit of length is the meter (m).
• The unit of mass is the kilogram (kg).
• The unit of time is the second (s).
• The unit of force is the newton (N), where:

${\displaystyle \ \!\mathrm {N} =1\ \!\mathrm {kg} \ {\frac {\mathrm {m} }{\mathrm {s} ^{2}}}\,\!}$

When working with units that are either large multiples or small fractions of these units, prefixes are often used in order to keep the numbers manageable. For example,

${\displaystyle 1000\ \!\mathrm {m} =1\ \!\mathrm {km} }$

The following table gives a more detailed description of prefixes.

Prefix Abbrev. Factor
peta- P 1015
tera- T 1012
giga- G 109
mega- M 106
kilo- k 103
hecto- h 102
deca- da 101
deci- d 10-1
centi- c 10-2
milli- m 10-3
micro- μ 10-6
nano- n 10-9
pico- p 10-12
femto- f 10-15
atto- a 10-18

All other measurements are derived using variations of these four basic units and the listed prefixes.

Common SI units are listed in the following table.

Value Units (long) Units (abbrev.)
Acceleration meter per second squared
${\displaystyle {\frac {\mathrm {m} }{\mathrm {s} ^{2}}}}$
${\displaystyle \!\mathrm {rad} }$
Angular Acceleration radian per second squared
${\displaystyle {\frac {\mathrm {rad} }{\mathrm {s} ^{2}}}}$
${\displaystyle {\frac {\mathrm {rad} }{\mathrm {s} }}}$
Area square meter
${\displaystyle \!\mathrm {m} ^{2}}$
Density kilogram per cubic meter
${\displaystyle {\frac {\mathrm {kg} }{\mathrm {m} ^{3}}}}$
Energy joule
${\displaystyle \!\mathrm {J} }$ or ${\displaystyle \!\mathrm {Nm} }$
Force newton
${\displaystyle \!\mathrm {N} }$
Frequency hertz
${\displaystyle \!\mathrm {Hz} }$ or ${\displaystyle {\frac {1}{\mathrm {s} }}}$
Impulse newton-second
${\displaystyle \!\mathrm {Ns} }$ or ${\displaystyle \!\mathrm {kg} {\frac {\mathrm {m} }{\mathrm {s} }}}$
Length meter
${\displaystyle \!\mathrm {m} }$
Mass kilogram
${\displaystyle \!\mathrm {kg} }$
Force Moment newton-meter
${\displaystyle \!\mathrm {Nm} }$
Power watt
${\displaystyle \!\mathrm {W} }$ or ${\displaystyle {\frac {\mathrm {J} }{\mathrm {s} }}}$
Pressure pascal
${\displaystyle \!\mathrm {Pa} }$ or ${\displaystyle {\frac {\mathrm {N} }{\mathrm {m} ^{2}}}}$
Stress pascal
${\displaystyle \!\mathrm {Pa} }$ or ${\displaystyle {\frac {\mathrm {N} }{\mathrm {m} ^{2}}}}$
Time second
${\displaystyle \!\mathrm {s} }$
Velocity meter per second
${\displaystyle {\frac {\mathrm {m} }{\mathrm {s} }}}$
Volume (solids) cubic meter
${\displaystyle \!\mathrm {m} ^{3}}$
Volume (liquids) litre
${\displaystyle \!\mathrm {L} }$ or ${\displaystyle \!\mathrm {dm} ^{3}}$
Work joule
${\displaystyle \!\mathrm {J} }$ or ${\displaystyle \!\mathrm {Nm} }$

## British and American Customary Units

While the International System of units is in common use throughout much of the world, engineers may still encounter British or American units. Therefore, it is a good idea to have some familiarity with them.

While the basic units in International System of units are length, mass, and time--with the unit of force defined in terms of these--in the British and American units, the base units are length, force and time, with mass being defined in terms of these.

• The unit of length is the foot (ft).
• The unit of force is the pound (lb), which is occasionally called pound-force (lbf).
• The unit of time is the second (s).

The unit of mass in British and American units is the slug. It is defined as the amount of mass accelerated at a rate of 1 ft/s^2 when 1 pound of force is applied.

${\displaystyle 1\ \!\mathrm {slug} =1\ \!\mathrm {lb} _{f}\ {\frac {\mathrm {ft} }{\mathrm {s} ^{2}}}}$

Occasionally, mass is described as a pound-mass. It is equal to the mass required to move one lb of weight when acted upon by the standard acceleration of gravity. On Earth, the standard acceleration of gravity is about 32.2 ft/s^2, this means that one slug is 32.2 lb(mass).

${\displaystyle 1\ \!\mathrm {slug} =32.2\ \!\mathrm {lb_{m}} }$

## Conversion from one System of Units to Another

While we can do all our calculations in one set of units or the other, as long as we are consistent, there are times we will want to convert from one system to the other.

• Unit of Length 1 ft = 0.3048 m
• Unit of Force 1 lb = 4.448 N
• Unit of Mass 1 slug = 1 lb-s^2/ft = 14.59 kg

As mentioned earlier, the second is the same in both systems of units and so no conversion is required.

Common British and American Customary units and their SI equivalents are listed in the table below.

Value Conversion
Acceleration
${\displaystyle 1\ {\frac {\!\mathrm {ft} }{\!\mathrm {s} ^{2}}}=0.3048\ {\frac {\!\mathrm {m} }{\!\mathrm {s} ^{2}}}}$
Area
${\displaystyle 1\ \!\mathrm {ft} ^{2}=0.0929\ \!\mathrm {m} ^{2}}$
Energy
${\displaystyle 1\ \!\mathrm {ft} \ \!\mathrm {lb} =1.356\ \!\mathrm {J} }$
Force
${\displaystyle 1\ \!\mathrm {lb} =4.448\ \!\mathrm {N} }$
Impulse
${\displaystyle 1\ \!\mathrm {lb} \ \!\mathrm {s} =4.448\ \!\mathrm {N} \ \!\mathrm {s} }$
Length
${\displaystyle 1\ \!\mathrm {ft} =0.3048\ \!\mathrm {m} }$
${\displaystyle 1\ \!\mathrm {mile} =1.609\ \!\mathrm {km} }$
Mass
${\displaystyle 1\ \!\mathrm {lb} \ \!\mathrm {mass} =0.4536\ \!\mathrm {kg} }$
${\displaystyle 1\ \!\mathrm {slug} =14.59\ \!\mathrm {kg} }$
${\displaystyle 1\ \!\mathrm {ton} =907.2\ \!\mathrm {kg} }$
Moment
${\displaystyle 1\ \!\mathrm {lb} \ \!\mathrm {ft} =1.356\ \!\mathrm {N} \ \!\mathrm {m} }$
Power
${\displaystyle 1\ \!\mathrm {ft} \ {\frac {\!\mathrm {lb} }{\!\mathrm {s} }}=1.356\ \!\mathrm {W} }$
${\displaystyle 1\ \!\mathrm {hp} =745.7\ \!\mathrm {W} }$
Pressure
${\displaystyle 1\ \!\mathrm {psi} =6.895\ \!\mathrm {kPa} }$
Stress
${\displaystyle 1\ \!\mathrm {psi} =6.895\ \!\mathrm {kPa} }$
Velocity
${\displaystyle 1\ {\frac {\!\mathrm {ft} }{\!\mathrm {s} }}=0.3048\ {\frac {\!\mathrm {m} }{\!\mathrm {s} }}}$
${\displaystyle 1\ \!\mathrm {mph} =1.609\ {\frac {\!\mathrm {km} }{\!\mathrm {hr} }}}$
Volume (solids)
${\displaystyle 1\ \!\mathrm {ft} ^{3}=0.02832\ \!\mathrm {m} ^{3}}$
Volume (liquids)
${\displaystyle 1\ \!\mathrm {gal} =3.785\ \!\mathrm {L} }$
Work
${\displaystyle 1\ \!\mathrm {ftlb} =1.356\ \!\mathrm {J} }$

### Example

According to the official National Hockey League rulebook, "The official size of the (hockey) rink shall be two hundred feet (200') long and eighty-five feet (85') wide." What are the dimensions in SI units?

#### Solution

From the above table:
${\displaystyle 1\ \mathrm {ft} =0.3048\ \mathrm {m} }$
Using dimensional analysis we find the length and width in meters.
${\displaystyle l\ =\ 200\ \mathrm {ft} \ \cdot \ {\frac {0.3048\ \mathrm {m} }{1\ \mathrm {ft} }}=\ 60.96\ \mathrm {m} }$
${\displaystyle w\ =\ 85\ \mathrm {ft} \ \cdot \ {\frac {0.3048\ \mathrm {m} }{1\ \mathrm {ft} }}=\ 25.96\ \mathrm {m} }$

## Significant Digits

When we talk about measurements and calculations, we need to understand the degree of accuracy involved.

The accuracy of our calculations cannot be more precise than the accuracy of our measurements.

Suppose we are provided with a distance to an accuracy of one decimal place, say 9.8 m. We are told an object travels this distance in 0.81 seconds. It does not make sense to say the object is traveling at a velocity of 12.11111111 m/s, that is, to eight decimal places.

This is because neither the distance nor the time taken to travel this distance is specified to this degree of precision. In fact, they are both specified to an accuracy of only two significant digits.

For reasons we will discuss shortly, we can say the object is traveling at a velocity of 12.1 m/s.

For many calculations in statics, we work to at most three significant digits.

### Rules for Finding the Correct Number of Significant Digits

In general, when making a calculation, the answer can not have more significant digits than any of the numbers used in calculating it. The number of significant digits in an answer is equal to the minimum number of significant digits used in the calculation.

Here are rules that will help outline whether or not a digit is significant or not.

1. Non-zero numbers are always significant.
2. Zeros placed in between two other digits are significant.
3. Zeros placed at the end of a number, after a decimal, are significant.

## References

1 - Both the principal SI units used in mechanics and the US Customary units and their SI equivalents are taken from Beer, Ferdinand P. and El Russell Johnston, Jr. "Vector Mechanics For Engineers, Statics" 3rd edition, McGraw Hill c 1977. It should be possible to find similar tables in other texts on this subject.

2 - Rules for taking significant digits are taken from www.physics.uoguelph.ca

 To do:I would like to add angle as a fundamental dimension. Define it as the ratio of the arc length to radius, in radians. We need this for rotational kinetics.

# Vector Math

Let's say you have a box on the ground, and the box is being pulled in two directions with a certain force. You can predict the motion of the box by finding the net force acting on the box. If each force vector (where the magnitude is the tension in the rope, and the direction is the direction that the rope is "pointing") can be measured, you can add these vectors to get the net force. There are two methods for adding vectors:

### Parallelogram Method

This is a graphical method for adding vectors. First, a little terminology:

• The tail of a vector is where it originates.
• The head of a vector is where it goes. The head is the end with the arrowhead.

This method is most easily executed using graph paper. Establish a rectangular coordinate system, and draw the first vector to scale with the tail at the origin. Then, draw the second vector (again, to scale) with its tail coincident with the head of the first vector. Then, the properties of the sum vector are as follows:

• The length of the sum vector is the distance measured from the origin to the head of the second vector.
• The direction of the sum vector is the angle.

#### Example

In the image at the right, the vectors (10, 53°07'48") and (10, 36°52'12") are being added graphically. The result is (19.80, 45°00'00"). (How did I measure out those angles so precisely? I did that on purpose.)

The native vector format for the parallelogram method is the 'polar form'.

### Computational Method

When you use the computational method, you must resolve each vector into its x- and y-components. Then, simply add the respective components.

#### Converting Polar Vectors to Rectangular Vectors

If a vector is given by (r, θ), where r is the length and θ is the direction,

• x = r cos θ
• y = r sin θ

#### Converting Rectangular Vectors to Polar Vectors

If a vector is given by ${\displaystyle \left\langle x,y\right\rangle }$,

• ${\displaystyle r={\sqrt {x^{2}+y^{2}}}}$
• ${\displaystyle \theta =\arctan {\frac {y}{x}}}$

Remember that the arctan() function only returns values in the range [-π/2, π/2]; therefore, if your vector is in the second or third quadrant, you will have to add π to whatever angle is returned from the arctan() function.

#### Example

Again referring to the image at the right, notice that the first vector can be expressed as ${\displaystyle \left\langle 6,8\right\rangle }$, and the second is equivalent to ${\displaystyle \left\langle 8,6\right\rangle }$. (Verify this.) Then, you simply add the components:

${\displaystyle \left\langle 6,8\right\rangle +\left\langle 8,6\right\rangle =\left\langle 6+8,8+6\right\rangle =\left\langle 14,14\right\rangle }$


You should verify that ${\displaystyle \left\langle 14,14\right\rangle }$ is equal to (19.80, 45°00'00").

## Multiplying Vectors

There are two ways to multiply vectors. I will not get into specific applications here; you will see many of those as you progress through the book.

### The Dot Product

The dot product of two vectors results in a scalar. The dot product is the sum of the product of the components. For example:

   < 1 , 2 >
∙ < 3 , 4 >
-----------
|   +-----> 2 x 4 = 8
+---------> 1 x 3 = 3
------
11


A useful relation between vectors, their lengths, and the angle between them is given by the definition of the dot product:

${\displaystyle {\vec {a}}\cdot {\vec {b}}=ab\cos \theta }$

• ${\displaystyle {\vec {a}}}$ and ${\displaystyle {\vec {b}}}$ are the vectors.
• ${\displaystyle a}$ and ${\displaystyle b}$ are the vectors' magnitude.
• ${\displaystyle \theta }$ is the angle between the vectors.

### The Cross Product

The cross product of two vectors results in another vector. The cross product is only applicable to 3-space vectors. Remember the three unit vectors:

• ${\displaystyle {\hat {i}}}$ is the unit vector along the x-axis
• ${\displaystyle {\hat {j}}}$ is the unit vector along the y-axis
• ${\displaystyle {\hat {k}}}$ is the unit vector along the z-axis

Now if you have two vectors ${\displaystyle {\vec {a}}=\left\langle x_{1},y_{1},z_{1}\right\rangle }$ and ${\displaystyle {\vec {b}}=\left\langle x_{2},y_{2},z_{2}\right\rangle }$, the cross product is given by solving a determinant as follows:

${\displaystyle {\vec {a}}\times {\vec {b}}={\begin{vmatrix}{\hat {i}}&{\hat {j}}&{\hat {k}}\\x_{1}&y_{1}&z_{1}\\x_{2}&y_{2}&z_{2}\end{vmatrix}}={\begin{vmatrix}y_{1}&z_{1}\\y_{2}&z_{2}\end{vmatrix}}{\hat {i}}+{\begin{vmatrix}z_{1}&x_{1}\\z_{2}&x_{2}\end{vmatrix}}{\hat {j}}+{\begin{vmatrix}x_{1}&y_{1}\\x_{2}&y_{2}\end{vmatrix}}{\hat {k}}}$


The cross product of two vectors, the lengths of those vectors, and the short angle between the vectors is given by the following relation:

${\displaystyle {\vec {a}}\times {\vec {b}}=ab\sin \theta }$


#### The Right-Hand Rule

Geometrically, the cross product gives a vector that is perpendicular to the two arguments. Notice the reference to a vector, not the vector. This is because there are infinitely many vectors that are normal to two non-zero vectors. The direction of the cross product can be determined using the right-hand rule: Extend the fingers of your right hand, lay your straightened hand along the first vector, pointing your finger tips in the same direction as the vector. Curl your fingers through the short angle from the first vector to the second vector. Your thumb will point in the direction of the product vector.

### Dots and Crosses of the Unit Vectors

#### Dot Products

• A unit vector dotted into itself gives one.
• A unit vector dotted into a different unit vector gives zero.

#### Cross Products

Order the unit vectors in this order: ${\displaystyle {\hat {i}},{\hat {j}},{\hat {k}}}$. Start at the first vector, move to the second vector, and keep going to the cross-product. If you moved immediately to the right first, the answer is positive. If you moved to the left first, the answer is negative. For example:

• ${\displaystyle {\hat {i}}\times {\hat {j}}={\hat {k}}}$
• ${\displaystyle {\hat {j}}\times {\hat {i}}=-{\hat {k}}}$

## Vector Rules

Given vectors ${\displaystyle {\vec {a}},{\vec {b}},{\vec {c}},}$ and scalar r:

• ${\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}}$
• ${\displaystyle r({\vec {a}}\cdot {\vec {b}})=(r{\vec {a}})\cdot {\vec {b}}={\vec {a}}\cdot (r{\vec {b}})}$
• ${\displaystyle {\vec {a}}\times {\vec {b}}=-{\vec {b}}\times {\vec {a}}}$
• ${\displaystyle r({\vec {a}}\times {\vec {b}})=(r{\vec {a}})\times {\vec {b}}={\vec {a}}\times (r{\vec {b}})}$
• ${\displaystyle ({\vec {a}}\times {\vec {b}})\cdot {\vec {c}}={\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}$