Introductory Dynamics

Front matter

Introductory Dynamics 2D Kinematics and Kinetics of Point Masses and Rigid Bodies Edition 1.1 Peter G. Steeneken

Delft University of Technology Introductory Dynamics 2D Kinematics and Kinetics of Point Masses and Rigid Bodies Peter G. Steeneken TU Delft Open Textbook Edition 1.1 - 2025

Title: Introductory Dynamics Subtitle: 2D Kinematics and Kinetics of Point Masses and Rigid Bodies Author: Peter G. Steeneken Publisher: TU Delft OPEN Publishing https://textbooks.open.tudelft.nl/textbooks Year of publication: 2025 Edition: 1.1 ISBN (paperback): 978-94-6366-794-4 ISBN (PDF): 978-94-6366-795-1 DOI: https://doi.org/10.59490/tb.83 DOI source files: https://doi.org/10.5281/zenodo.14598784 Cover image: Peter G. Steeneken Cite as: P.G. Steeneken, Introductory Dynamics, Ed. 1.1, TU Delft Open Publishing, Delft, 2025.

Part I Introduction

1 Preface This textbook provides you with an overview of the methods to analyse and predict the dynamic motion of point masses and rigid bodies in the twodimensional plane based on Newton’s laws. The book is aimed at first-year bachelor students in science and engineering. The recommended prerequisite knowledge for the textbook includes high-school physics, basic calculus (differentiation and integration), vector algebra, linear ordinary differential equations, and knowledge of statics. Furthermore, the gradient is introduced from vector calculus. Most concepts are briefly introduced, such that this textbook might in principle also be used by readers without this prerequisite knowledge. In case you spot errors, inaccuracies or if you have other suggestions to improve this textbook it would be highly appreciated if you send your feedback to p.g.steeneken@tudelft.nl. 1.1 Guidelines for studying dynamics This textbook aims to bring you basic understanding of dynamics and provide an engineering toolbox by which you can solve many problems in dynamics, both analytically and numerically. However, it is important to learn how to use this toolbox to apply it in new situations. The only way to learn this well is to solve problems. For this reason, it is very important, after having studied a part of the textbook, to practice solving problems yourself. Always first try to solve the problem yourself before looking at the solution, since you learn much more by finding the solution yourself, even if that costs more time. Getting stuck in solving a problem is probably the experience from which you learn the most. It points out what parts you don’t understand well enough, encourages you to study those parts better by reading the book or looking at similar examples, and after solving the problem the experience will ensure you will not easily forget how to solve similar problems. So, don’t give up too early or look at the solution, if you persist you often will manage. Besides teaching you the basics of dynamics, an important aim of this textbook is to teach you to get rid of ’bad habits’ you might have acquired in high school by implementing the following guidelines:

14 1 PREF ACE 1. Work with equations instead of filling in numerical values while analysing a problem. The aim is to learn how to solve problems with equations instead of filling in numbers at an early stage. Not only are you more likely to make errors when filling in numbers, it is also much harder to detect those errors afterwards (e.g. by checking units). Moreover, when having an equation as solution, it has a larger degree of validity and can be used to predict trends and optimise designs. Only fill in numbers after having reached the final equation. 2. Work with vectors instead of scalars. A position, force, moment, velocity or acceleration in 2 or 3 dimensional space is always a vector. Therefore you should initially always use vector notation in 2D and 3D , and only work with scalar values when appropriate, e.g. after having projected the vectors on a coordinate axis. Correct vector usage and notation will become increasingly important when you progress with your studies, so best get used to it early. Note that most problems in this book will focus on motion in the 2-dimensional xy-plane. Nevertheless, even in that case moment and angular velocity vectors point along the z direction. Moreover, most vector equations that are presented in this textbook will still be valid in 3D, unless they have a subscript 2D. 3. Using a structured approach instead of jumping to conclusions too soon. In the analysis of dynamic systems, we will not assume anything, but use structured procedures to solve problems. For example, if a mass moves over a surface, we will not jump to the conclusion that the upward normal force equals the weight of the mass (which is not always the case for a curved surface), but determine the normal force from Newton’s laws. 4. Solve problems based on understanding, instead of by memorising example problems, and copying the methodology without understanding how it works. Although this memorising and solving many standard problems might seem effective for passing exams, it will leave you helpless when you encounter a problem that you did not see before. Therefore, focus on building understanding and ask questions if things are not clear.

This textbook aims to form a bridge between high-school physics and academic literature, providing more rigorousness and insight into the material. Available textbooks are often either too advanced and complex, or do not offer sufficient depth, insight or rigorousness in my view. This textbook provides a solid basis in dynamics of which I hope that it will help you to bridge the gap towards understanding more advanced academic books and articles.

1.2 TOPICS 15 1.2 Topics The outline of the topics that will be discussed in this textbook is as follows: 1. Introduction, elemental concepts, forces, vectors and mathematics 2. Dynamics of point masses a) Kinematics b) Kinetics c) Work and energy d) Impulse and momentum 3. Dynamics of rigid bodies a) Kinematics b) Kinetics c) Work and energy d) Angular impulse and momentum 4. Vibrations 5. Solution strategy dynamics Although all equations and theory will be derived and presented for motion in 3D space (unless specifically indicated otherwise for special cases), most examples and problems will focus on point masses and rigid bodies moving in a 2D plane. The dynamics of deformable objects will not be part of the textbook. Several section headings are marked with a * at the end of the section title, to indicate that the main text can be followed without reading these parts because they contain derivations or background information. The following introductory chapters might be skipped if they are already familiar. The core of the textbook starts in Ch. 5. 1.3 Text structure In this textbook, the techniques and models that scientists and engineers need to design and analyse dynamical systems will be introduced step by step, starting from Newton’s laws. Eventually at the end of the textbook we will have discussed a toolbox of equations and techniques that can be valuable for your future career.

The text consists of different parts: 1. Descriptions and explanations 2. Derivations of key equations and concepts

16 1 PREF ACE 3. Concepts and techniques in dynamics 4. Practical examples, and sample problems that illustrate the use of these techniques Although derivations of equations might be skipped without following the main line of the textbook, it is highly recommended to read them, since they do provide more insight into the theory and demonstrate beautifully how Newton’s laws, despite their simplicity, can account for a tremendous set of complex phenomena in dynamics.

To highlight important concepts, they are introduced like this: Concept. Principle of linear impulse and momentum ∑ j mj⃗vj (t1) + ∑ i,j ∫ t2 t1 ⃗Fi,jdt =∑ j mj⃗vj (t2) (1.1) Similarly, derivations are marked like this: Derivation. We will here derive the principle of impulse and momentum from Newton’s laws.

Examples to illustrate how to apply the concepts and techniques are provided in an example environment, with the letter in front of the example indicating the difficulty level (S: simple, M: medium and A: advanced). S Example 1.1 We consider the following example situation. Exemplary solution The dynamics of this system can be solved as follows. Problems are included in these notes as follows: M Problem 1.1 This is a test problem. 1.4 Changes in edition 1.1 During the first year of using edition 1.0 of this book several additions and corrections have been implemented. Thanks to students and colleagues that suggested improvements! For this reason we release edition 1.1 of the textbook. Here we provide a list of the most important changes implemented in edition 1.1. • Subsection Sec. 2.1.3 was added, with Table 2.2, which provides an overview of the symbols and subscripts used in this textbook. Some

1.5 ACKNOWLEDGEMENTS 17 students have indicated that they find symbols with subscripts complicated or intimidating. In this section the motivation for the notation with subscripts is given. The convention we adopt is that a vector ⃗F always represents a force, and the subscripts indicate the type of force. This provides clarity in contrast to other textbooks where vectors like ⃗N, ⃗W, and ⃗P are used for forces. • In Ch. 7 the terms energy was replaced by the term mechanical energy to indicate the sum of kinetic and potential energy. The motivation for this change is to make it more clear that energy is always conserved in a closed system. On the contrary, the mechanical energy of a closed system is only conserved in a closed system if no work is performed by non-conservative forces, because non-conservative forces can convert mechanical energy into thermal energy (or heat). • Fig. 6.6 was added and more explanation on how to project tnb coordinates on Cartesian and cylindrical coordinates is provided. • A few errata were found, the most important are: a factor c22 was missing in Example 5.3. A division sign in the equation⃗FB =mB⃗aB was removed in Sec. 6.12. There was a typing error in variable Vg,tot in Eq. (7.43). The description of the static equilibrium yA,st in Fig. 13.1 was not completely correct. • An index was added. 1.5 Acknowledgements In 2021 I started writing this textbook to support the course Introductory Dynamics that I teach for first year BSc. students, motivated by Heike Vallery who wrote a book on Advanced Dynamics [ 9] and also provided me with a template [ 9] and contributed parts of her notes that have been reused in adapted form in Ch. 2 and Ch. 3. I would like to thank Jaap Meijaard for thorough review of the full text. I also would like to thank Heike Vallery, Arno Stienen, Niels Bouman and Boris Ullrich, for their help and feedback on the manuscript. I also acknowledge the support from the department of Precision and Microsystems Engineering and the Mechanical Engineering faculty of TU Delft in writing this textbook. TU Delft Open Textbooks has helped me to publish the book. I provide it with c b licence, with the Latex source code and images on Zenodo (see DOI on page 2), in order to facilitate distribution and education, and to enable others to reuse it and expand on it, which is encouraged and permitted as long as a reference to this textbook is provided. Note that none of the underlying dynamics theory presented in this textbook

18 1 PREF ACE is new. All of it is based on beautiful scientific work that was developed over several centuries by many scientists and educators, most notably Newton [ 7, 6] and Euler [3].

2 Notation, Math, and Engineering Basics This chapter will discuss notation conventions, unit usage, significant digits and repeat mathematical background knowledge that will be needed to follow this textbook. 2.1 Notation In this section we discuss the used notation, labelling and typesetting in this textbook. Sec. 2.1 and Sec. 2.2 are an adaptation of the text of Vallery and Schwab [9]. 2.1.1 Typesetting of scalars, vectors, and matrices There are several ways to typeset scalars, vectors, unit vectors, and matrices as depicted in Table 2.1. In particular for vectors and unit vectors there are multiple methods applied in literature. Although in many works vectors are only written with boldface italics, it is hard to clearly write a boldface letter in handwriting. Therefore we here use a combination of boldface and vector arrow above the letter for clarity and similarity to the handwritten vector. Similarly, for unit vectors (vectors with a magnitude of 1), we use a boldface letter with a hat ˆ above it. All variables are italics, also if they appear in subscripts or superscripts. Matrices are non-italic and also names and texts in sub or superscripts are non-italic except if they are single letters[8]. Table 2.1: Notation of scalars, vectors, unit vectors and matrices This book Other notations Scalars F Vectors ⃗F ⃗F , F ,F , F Unit vectors ˆı, ˆρ ux, e1, i, ˆx Matrices R R

20 2 NOTATION, MATH, AND ENGINEERING BASICS 2.1.2 Quantities, labels and subscripts In this textbook we generally follow the guidelines set out in the Red Book [2]. To label objects or points we either use letters like A,B,... , numbers i = 1, 2,... or short words like rope or ball. For adding extra information and distinguishing quantities, we can add subscripts like in mrope and mball. Multiple labels can be separated in the subscript by commas. Italic integer variables, likei in Fi (with i = 1, 2,... ) can indicate that multiple quantities F1,F 2,... exist, each with a different value ofi. Although labels are not always essential, make sure to always use sufficient labels to uniquely identify each quantity. So, if there is only one mass in your problem, it is fine to just m to identify it. However, if there are two point-masses, subscripts mA and mB are needed for unique identification. With an axis coordinate as a subscript, like x in Fx, we indicate a projection of a vector ⃗F along the x-axis. A special notation for relative position, velocity and acceleration vectors is used (see also Sec. 3.1.2). A position vector that points to a point A from a point P will be denoted as ⃗rA/P =⃗rA−⃗r/P , which reads: “position of A with respect to P ” . Analogous notation will be used for other relative quantities, such as angles, velocities and accelerations (see Sec. 3.1). For components of 3-dimensional vectors, there are three possible ways of notation of which we will usually use the last one because it is the shortest to write: ⃗r =  rxry rz   =(rx ry rz)T =rxˆı +ryˆȷ +rzˆk. (2.1) The superscript T indicates the transpose of the vector (or matrix), and is sometimes used to convert the column vector to a row vector to save space. For components of m-dimensional vectors, one can also use the index notation , with unit vectors written like ˆe1 = ˆı, which has the advantage that it can be written even more compactly as a sum: ⃗r =   r1 r2 ... rm   =(r1 r2 ... r m)T = m∑i=1riˆei. (2.2) Elements ofm×n-dimensional matrices receive two indices, for row and column: A =   a11 a12 ... a 1n a21 a22 ... a 2n am1 am2 ... a mn  . (2.3)

2.2 WORKING WITH UNITS 21 Note that in dynamics, we use Newton’s notation (also called dot notation or fluxions) for time derivatives, indicating them by a dot above the variable: ˙x = dxdt (2.4) ¨x = d2xdt2 (2.5) In mathematics the prime symbol ′ is often used for derivatives like f′ = dfdx. In dynamics it is advisable not to use the prime symbol to avoid confusion, and instead clearly indicate with respect to what variable the derivative is taken e.g. df dx or dfds . Also clearly indicate if it is a total derivative dfdx or partial derivative ∂f ∂x . In this textbook the prime symbol ′ will not be used for derivatives but is sometimes used as a label. Finally, when making multiplications of scalar quantities, both· and× are used, such that a·b =a×b. For vectors these symbols indicate the dot and cross product which clearly are different: ⃗a·⃗b̸=⃗a×⃗b. 2.1.3 Overview of symbols and subscripts If you are not used to them, the mathematical symbols and names for points, objects, coordinates, scalar quantities and vector quantities, having sometimes multiple subscripts, might look complicated. However, actually one can get used to them quite quickly, since the number of different symbols used in this book is quite limited, as can be seen in Table 2.2, which contains almost all symbols that you will encounter in this textbook. The large letter indicates the type of quantity or coordinate, while the subscripts give information on the point(s), time(s), or object that it relates to. So the symbols provide a very compact way to write down dynamical systems in a precise mathematical way, while also providing a connection between drawings and mathematical equations. At this moment the symbols in Table 2.2 have not been explained yet, but in the rest of this textbook we will clarify the symbols in Table 2.2 in more detail and discuss them again in Sec. 10.7.2. 2.2 Working with units There are several rules when working with units that are not always obvious, and sometimes not correctly applied, even in standard textbooks. Errors with units have led to serious confusions and accidents like the collapse of a Mars orbiter and an aeroplane running out of fuel in mid-air. It is therefore

22 2 NOTATION, MATH, AND ENGINEERING BASICS Table 2.2: Important symbols and notation of subscripts in dynamics Symbol Represents A Name of point (mass) or object mA Mass of object A t1 Specific time t1 on the time axis t sA Position coordinate of A on the path axis/curve s k Spring constant k of spring k g Gravitational constant g = 9.81 m/s2 ⃗rA Position vector of A ⃗rA/B Relative position vector of A with respect to point B ⃗FA Force acting on A FAx x-component of vector ⃗FA ⃗MA/G Moment on A relative to the CoM G of the system ⃗pG Momentum of centre of mass G of a system IC/G Moment of inertia of C relative to the center of mass G ⃗LC/P,2D Angular momentum of C relative to point P , equation only valid for planar 2D motion x,y,z , ρ,ϕ,z Cartesian and cylindrical coordinates ˆı,ˆȷ, ˆk, ˆρ, ˆϕ, ˆk Cartesian and cylindrical unit vectors ˆt, ˆn, ˆb Unit vectors for natural coordinates ⃗r,⃗v,⃗a,⃗ω,⃗α Main kinematic vectors W,P,T,V Work, power, kinetic and potential energy ⃗Ji,12 Impulse on point mass i during time-intervalt1-t2 ⃗HC,12 Angular impulse on object C during time-intervalt1-t2

2.2 WORKING WITH UNITS 23 important for every engineer to be able to correctly work with units and we repeat the most important rules here.

In mechanics, we often deal with scalar variables that represent physical quantities that have a measurement unit, like forces, velocities, or positions. Scalar variables consist of the product of two elements: a number and a measurement unit.

S Example 2.1 Force can be expressed as the product of a numerical value and the measurement unit newton: F = 3· N = 3 N.

Following the ISO norm [ 5] and the “Red Book” [ 2], for quantities that have a measurement unit, the numerical value of a scalar variable Q that represents such quantities is denoted by {Q}, and the measurement unit is denoted by [Q], such that Q ={Q}· [Q]. (2.6) S Example 2.2 A mass m = 3.5 kg has the numerical value{m} = 3.5 and the unit [m] = kg. The same quantity value can be expressed as 3500 g. Then, the numerical value is{m} = 3500, and [m] = g.

Note that square brackets only have meaning when placed around the physical quantity. For axes labels one commonly uses round brackets [ 4, 8] around the unit like for a time axis: t (s). Another option [ 1] is to use division: t/s. Generally, the letters of units are printed in roman (upright), those of scalar variables in italic.

S Example 2.3 For timet, good use is: t = 4 s,t = 4 (s),t/s = 4, or [t] = s, not good use is: XXXXt = 4 [s], or ZZt [s]. Besides paying attention to the units, also youneed to make sure that if you have a vector on the left side of the = sign, you also need to have a vector on the right side.

From Eq. (2.6) it can be seen that every scalar equation in mechanics can be split into two equations, one for the numerical values, and one for the units. For example, an equation likeF =ma, withF = 1 N,m = 1 kg anda = 1 m/s2 can be split into: F = m·a (2.7) {F}[F ] = {m}[m]·{a}[a] (2.8) {F} = {m}·{ a}→ 1 = 1· 1 (2.9) [F ] = [ m]· [a]→ N = kg· m/s2 (2.10) When making a calculation it is important that in the end result both the numerical values and the units are correct since obviously, in equations, the

24 2 NOTATION, MATH, AND ENGINEERING BASICS units on both sides of the equal sign need to be identical, otherwise the equality can never hold. Normally, the units automatically work out well if you work with SI units. In some cases, one has to deal with equations like a =c1t2. In that case the unit of the constant c1 is not given, but can be obtained from the equation [a] = [c1][t2], which can be written as m/s2 = [c1]· s2, from which it follows that the unit of c1 is: [c1] = m/s4.

A general advice is to work as long as possible with variables, and only to substitute numbers and units in the final solution. Units almost always help as a part of plausibility checks for calculations. Moreover, writing the numerical value and unit in each step of the derivation is much more work, and errors in numerical values are harder to find than in equations. In this textbook we mostly work with the International System of Units, also called SI units, the SI system, or metric units, which are the world’s most widely used system of measurement. If you have to work with non-SI units, as a habit first convert it to the SI unit as a first step to prevent problems later on. Often this conversion is just a matter of multiplying the non-SI unit value by a constant factor, e.g. 1 .0 inch = 1.0 inch× 0.0254 m/inch, however in some cases the conversion is more complex, like when converting temperature in Fahrenheit to degrees Kelvin. Preferably also work in the base SI unit, remove any factors related to prefixes (like converting mm to m by dividing by a factor 1000) before starting calculations, using scientific notation (e.g. 3 .02× 108) for very large or small values.

There are 7 base SI units, and the only base SI units relevant in this textbook are m, s and kg. There are 22 derived SI units of which we use 6 in this textbook: the rad = m/m, Hz = 1/s, N = kg m/s2, Pa = N/m2, J = Nm, and W = J/s. The unit to use for a certain quantity or parameter can often be determined by evaluating known dynamics equations. As an example, since x has unit m and t has unit s, we can use the equation for acceleration a = d2xdt2 to determine that acceleration a has the unit m/s2 (note how the square sign 2 is placed behind thet but before the x in the second derivative notation). Similarly, one obtains from F =m×a that 1 N = 1kg× 1 m/s2 = 1 kg m/s2. This method can often help to find the right unit by using a known equation, such that units don’t have to be memorised. Keep in mind that the equations in this textbook only work correctly if one uses SI units or units derived from SI units (like N or W). In some cases the scalar variable represents a unitless quantity. Examples of unitless quantities are integer numbers (for example the number of wheels under a car) or ratios of quantities with the same unit (for example the ratio of two distances). Most mathematical functions, like trigonometric or logarithmic functions, only accept unitless quantities. For example there is no mathematical way to calculate tan(1 m). So, always make sure only unitless numbers or ratios

2.2 WORKING WITH UNITS 25 are used. E.g. log(x2/x1) is fine, but log(x2) cannot be calculated if [ x2] = m. It is important to note that additions and subtractions can only be performed between scalar quantities of the same unit, while multiplications, divisions and powers can be taken on quantities of different units as long as the operation is also applied to the respective units.

A special case are angles, even though they usually have degrees ◦ or rad behind them, these angle designations are actually unitless numbers, which measure e.g. the ratio of the length of a circular arc divided by the length of the perimeter of a full circle. The degree or radian symbol only indicates how the trigonometric function should be applied. When not otherwise specified in this textbook we work with radians. Since angles are unitless, most mathematical functions can accept angles as an argument.

When dealing with vectors, one also needs to specify the direction of the vector using one or more unit vectors, so for specifying a vector one uses the product of a number, a measurement unit and at least one unit vector. Subtraction and addition is only possible for vectors with the same unit, scalar products are possible as long as the units are also multiplied, so for instance ⃗v×t = ∆⃗x is a correct equation, because m/s× s = m and because quantities on both sides of the = sign are vectors.

Expressions and equations may contain mixtures of quantities given with their numerical value and quantities given as variables. In all cases one needs to ensure that the units on both sides of the equal sign = are equal, and that one cannot equate a scalar to a vector or matrix. If a vector quantity evaluates to zero one we use the zero vector ⃗0, which is a vector with magnitude zero and no direction.

S Example 2.4 For a particle of mass m = 3 kg having acceleration vector ⃗a = 1 m/s2ˆı, the resultant force vector ⃗F on the particle can be calculated as: Correct is: ⃗F =m⃗a = 3 kg× 1 m/s2ˆı = 3 kg m/s2ˆı = 3 Nˆı. Incorrect is: (((((( hhhhhh⃗F = 3·⃗a N, because the vector ⃗a has unit m/s2 such that the units on both side of the equal sign are different.

S Problem 2.1 A particle performs a harmonic oscillation, such that its location coordinate x is a function of amplitude x0, angular frequency Ω, and time t: x =x0 sin(Ωt). (2.11) Fill in suitable units for x, x0, Ω, and t and check the equation for consistency. S Example 2.5 Which of these expressions make correct use of units, for mass m, time t, force F , angular velocity ω, and angle θ? Correct the mistakes.

26 2 NOTATION, MATH, AND ENGINEERING BASICS A. m = 20 B. m = 3 N C. F = 3·m [N] D. F = 20 N E. [ m] = kg F. m/[m] = 20 G. t = 3 s H. ω = 2 rad/s I. θ =ω·t = 2·t [s] J. {m} = 20 K. ω =t2 L. ω = 1 rad/s·t2 M. F (t) = cos(1 rad/s·t) N N. F (t) = 3e−t N O. F (t) = ln(t2/s2) kg· m/s2 Exemplary solution m̸= 20 unit omitted; m = 20 kg m̸= 3 N wrong unit for mass quantity; m = 3 kg F̸= 3·m [N] unit omitted and unit inserted with wrongly used square brackets; F = 3 m/s2·m F̸= 20N roman variable, italic unit symbol; F = 20 N [m] = kg correct! m/[m] = 20 correct! t = 3 s correct! ω = 2 rad/s correct! θ =ω·t̸= 2·t [s] wrong unit and [s] is wrong;θ = 2 rad/s·t {m} = 20 correct! ω̸=t2 unit mismatch; e.g. ω = 1 rad/s3·t2 ω̸= 1 rad/s·t2 unit mismatch; e.g. ω = 1 rad/s3·t2 F (t) = cos(1 rad/s·t) N correct !

F (t)̸= 3e−t N exponent should not have units; e.g. F (t) =e−t/s N F (t) = ln(t2/s2) kg· m/s2 correct ! 2.3 Precision, significant digits and errors When designing new systems, it is essential to be sure that they satisfy the required specifications. Therefore, when working with numerical values, it is essential to be aware of the precision of these numbers. Neglecting the consideration of the precision can lead to big problems, like parts of machines

2.3 PRECISION, SIGNIFICANT DIGITS AND ERRORS 27 that do not fit together. Uncertainties in quantities with respect to the average value can occur both due to actual variations (in time) in the quantity that is measured and due to uncertainties in the measurement technique that is used to determine the quantity. 2.3.1 Precision and significant digits The most common method to determine the uncertainty in a quantity is to measure the quantity q a large number of times and determine the average valueqavg and standard deviation σq. For example we measure the diameter of a 1 euro cent coin N times with a calliper, analyse the results and obtain d =davg±σd = (16.20± 0.05) mm. The standard deviation σd is a measure of the uncertainty (imprecision) in a single measurement 1 of the value of the coin diameter d. If many different euro cent coins are measured variations between the euro cent coins can also increase the uncertainty, thus both measurement uncertainty and object variations can contribute to the total uncertainty. Assuming a Gaussian distribution of the measurement values, the value of the standard deviation gives a probability of 68% that the diameter of the coin is between 16.15 mm and 16.25 mm.