10.7: Vectors in dynamics

In this textbook many different kinetic and kinematic vectors are introduced. We summarise these vectors, their labelling and their dependence on reference points.

10.7.1 Dependencies of vectors on reference points

When a single force \(\overrightarrow{\boldsymbol{F}}_{A}\) acts on a rigid body \(C\) with point of action \(\overrightarrow{\boldsymbol{r}}_{A}\) and reference point \(\overrightarrow{\boldsymbol{r}}_{P}\) for rotations, a variety of vectors are used to analyse the dynamics. These vectors and their dependence on position vectors \(\overrightarrow{\boldsymbol{r}}_{A}\) and \(\overrightarrow{\boldsymbol{r}}_{P}\) are summarised in Table 10.1.

Table 10.1: Dependencies of the vectors used to analyse the dynamics of a rigid body \(C\) with CoM \(G\), on the point of action \(A\) of force \(\overrightarrow{\boldsymbol{F}}_{A}\) and the choice of reference point \(P\).

	\(\overrightarrow{\vec{r}}_{A}\)	\(\overrightarrow{\boldsymbol{r}}_{P}\)
\(\overrightarrow{\boldsymbol{F}}_{A}\)	\(\checkmark\)	\(x\)
\(\overrightarrow{\boldsymbol{a}}_{G}\)	\(x\)	\(x\)
\(\vec{M}_{A / P}\)	\(\checkmark\)	\(\checkmark\)
\(\vec{M}_{\text {couple }}\)	\(\overrightarrow{\boldsymbol{r}}_{A / B}\)	\(x\)
\(\overrightarrow{\boldsymbol{\alpha}}_{C}\)	\(\checkmark\)	\(x\)
\(\overrightarrow{\boldsymbol{L}}_{C / P}\)	\(x\)	\(\checkmark\)
\(\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{L}}_{C / P}\)	\(\checkmark\)	\(\checkmark\)

Note that the subscript \(P\) is added to the vectors \(\overrightarrow{\boldsymbol{M}}_{A / P}\) and \(\overrightarrow{\boldsymbol{L}}_{C / P}\) because they depend on \(\overrightarrow{\boldsymbol{r}}_{P}\). It might seem strange that a moment vector can depend on the choice of the reference point \(P\), while the dynamics of the system (i.e. the acceleration \(\overrightarrow{\boldsymbol{a}}_{G}\) and angular acceleration \(\overrightarrow{\boldsymbol{\alpha}}_{C}\) ) do not depend on it. To understand this, we repeat Euler’s second law for planar motion Equation 10.22:

\[M_{C / P, 2 D} \hat{\boldsymbol{k}}=\overrightarrow{\boldsymbol{r}}_{G / P} \times\left(m_{\mathrm{tot}} \overrightarrow{\boldsymbol{a}}_{G}\right)+I_{C / G, z z} \alpha_{C} \hat{\boldsymbol{k}} \tag{10.57} \label{10.57}\]

It can be seen that changing \(P\) only results in a change in the relative position \(\overrightarrow{\boldsymbol{r}}_{G / P}\), such that even for identical dynamics ( \(\overrightarrow{\boldsymbol{a}}_{G}\) and \(\alpha_{C}\) both constant), the moment vector \(\overrightarrow{\boldsymbol{M}}_{C / P, 2 D}\) acting on the system \(C\) can change. Another way to look at is to realise that Equation 10.22 shows that a moment vector \(\overrightarrow{\boldsymbol{M}}_{C / P}\), that originates from a force \(\overrightarrow{\boldsymbol{F}}_{A}=m_{\mathrm{tot}} \overrightarrow{\boldsymbol{a}}_{G}\), has two effects: 1. Linear acceleration of the CoM of the system, such that the angular momentum \(\overrightarrow{\boldsymbol{L}}_{C / P}\) of \(G\) with respect to \(P\) changes. 2. Generate an angular acceleration \(\overrightarrow{\boldsymbol{\alpha}}_{C}\). Only the first
effect depends on the choice of \(P\). Remember that the reference point \(P\) should either be chosen as a fixed point in an IRF, or as the (accelerating) point \(P=G\) that moves along with the CoM of the system \(C\). It is not allowed to choose another accelerating or rotating point as reference point \(P\), since then Euler’s second law does not hold anymore. Finally note that the dependence of \(\overrightarrow{\boldsymbol{\alpha}}_{C}\) and \(\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{L}}_{C / P}\) on the point of action \(A\) of the force in Table 10.1 is not a direct dependence, since you can calculate these vectors from kinematics without knowing what \(A\) is (there is only a dependence on the point of action via Euler’s second law), therefore \(A\) is not needed as subscript in these cases.

10.7.2 Labelling scalars and vectors in dynamics

As discussed in the previous subsection, the magnitude and direction of vectors can depend on the point of reference and point of action, and therefore these points have to be clearly identified to prevent making mistakes. Here we will describe how to provide this kind of important information via labels (see also Sec. 2.1.2). Labelling can save time, since it avoids having to writing for each symbol a sentence like: \(" \overrightarrow{\boldsymbol{M}}_{A / P}\) is the moment vector generated by force \(\overrightarrow{\boldsymbol{F}}_{A}\) (with point of action \(A\) ) with respect to reference point \(P "\). Labels reduce the risk that confusion can occur about the meaning of a symbol. Here we provide a few conventions we recommend for using labels in dynamics:

1. Every mathematical quantity (e.g. scalar, vector, tensor) should get a unique symbol to identify it, conventions are:

• \(\overrightarrow{\boldsymbol{r}}, \overrightarrow{\boldsymbol{v}}, \overrightarrow{\boldsymbol{a}}\) are symbols for position, velocity and acceleration vectors.
• \(\overrightarrow{\boldsymbol{\omega}}, \overrightarrow{\boldsymbol{\alpha}}\) are symbols for angular velocity and acceleration vectors.
• \(m, I, \overrightarrow{\boldsymbol{p}}\) and \(\overrightarrow{\boldsymbol{L}}\) are mass, moment of inertia, momentum and angular momentum vectors.
• \(\overrightarrow{\boldsymbol{F}}\) and \(\overrightarrow{\boldsymbol{M}}\) are force and moment vectors (so e.g. don’t use \(\overrightarrow{\boldsymbol{N}}\) for a normal force but instead use \(\overrightarrow{\boldsymbol{F}}_{\mathrm{N}}\) ).
• \(W, V\) and \(T\) are work, potential energy and kinetic energy.
• \(\overrightarrow{\boldsymbol{J}}\) and \(\overrightarrow{\boldsymbol{H}}\) are impulse and angular impulse.

2. If the problem deals with multiple objects, each object should get a unique label (letter, number, word), that is indicated in the sketch.
3. Every relevant point (e.g. position, CoM, point of action, reference point) in the sketch should get a unique label.
4. Every quantity should get sufficient subscripts to define it uniquely. We will use the following conventions for these definitions:

• \(\overrightarrow{\boldsymbol{r}}_{A}, \overrightarrow{\boldsymbol{v}}_{A}, \overrightarrow{\boldsymbol{a}}_{A}\) get a subscript \(A\) to indicate the point or point mass \(A\) to which they refer.
• \(\overrightarrow{\boldsymbol{\omega}}_{C}, \overrightarrow{\boldsymbol{\alpha}}_{C}\) get subscripts \(C\) to indicate the system/rigid body \(C\) they refer to.
• Each mass \(m_{C}\) gets a subscript to identify the object \(C\) it refers to, while a moment of inertia \(I_{C / P, z z}\) also refers to the reference point \(P\). The component \(z z\) of the inertia tensor can be left out if it is clear that one deals with planar motion in the \(x y\)-plane.
• \(\overrightarrow{\boldsymbol{p}}_{C}=m_{C} \overrightarrow{\boldsymbol{v}}_{C, G}\) is the momentum of object \(C\) with total mass \(m_{C}\) and \(\overrightarrow{\boldsymbol{L}}_{C / P}\) is its angular momentum vector with respect to reference point \(P\).
• \(\overrightarrow{\boldsymbol{F}}_{A}\) is a force that acts on point of action \(A\). A label to indicate the source of the force can be added, like \(\overrightarrow{\boldsymbol{F}}_{\text {rope }, A}\). If two objects \(i\) and \(j\) generate forces on each other, then \(\overrightarrow{\boldsymbol{F}}_{i j}\) is the force acting on \(i\) and \(\overrightarrow{\boldsymbol{F}}_{j i}\) is the force acting on \(j\), with \(\overrightarrow{\boldsymbol{F}}_{i j}=-\overrightarrow{\boldsymbol{F}}_{j i}\) according to Newton’s third law.
• \(\overrightarrow{\boldsymbol{M}}_{A / P}\) is the moment vector as a result of the force \(\overrightarrow{\boldsymbol{F}}_{A}\) with respect to reference point \(P\). Note that we use the slash / with the meaning ’with respect to’ or ’relative to’ similar as in relative position vectors.
• \(\overrightarrow{\boldsymbol{J}}_{A, 12}\) is the impulse vector of a force \(A\) on the time interval between \(t_{1}\) and \(t_{2}\) and \(\overrightarrow{\boldsymbol{H}}_{A / P, 12}\) is the angular impulse vector of a force \(A\) with respect to reference point \(P\) between \(t_{1}\) and \(t_{2}\).
• \(W_{A, 12}\) is the work done by a force \(A\) between two states (times or positions), \(V_{A}\) is the potential energy of a force (field) \(A\) and \(T_{C}\) is the kinetic energy of an object \(C\).

5. In some cases quantities can be labelled uniquely with fewer subscripts. E.g. if there is only one object with mass in the problem, that mass can be given the label \(m\) without subscript, but it is never allowed to leave out the reference point labels \(P\) for moments \(\vec{M}_{A / P}\), moments of inertia \(I_{P, z z}\), angular momentum \(\overrightarrow{\boldsymbol{L}}_{P}\), and impulse vectors \(\overrightarrow{\boldsymbol{H}}_{A / P, 12}\), since the whole meaning of those quantities is tied to the choice of reference point \(P\).
6. The use of labels according to the conventions above is a replacement for clearly defining every symbol in words or equations separately. Providing such a definition is also a valid approach.