Mass/Inertia Scalars#

  • Mass: measure of amount of materia in a body. Units of measurement: \(kg\), \(lb\).

  • Mass center: consider a set of particles as shown below which together make up the system of particles \(S\):

The \(i^{\text{th}}\) particle has a mass \(m_i\)

../_images/110.png

\(S^*\) is a fictitious particle such that:

(25)#\[\sum_{i} m_i {\bf p}_i = 0\]

This fictitious particle is called mass center. So, how does one locate the mass centre from a point \(O\)?

../_images/35.png

  • \({\bf r}^*\); position vector from \(O\) to \(S^*\)

  • \({\bf q}_n\); position vector from \(O\) to \(P_n\)

So, the position vector to locate the \(n^{th}\) particle is:

\[ {\bf p}_n = {\bf r}^* - {\bf q}_n \]

Thus, expanding (25)

../_images/41.png

For a continuum:

../_images/51.png
../_images/61.png

where, \(dm\) is elemental mass that can be obtained from density \(\rho\) and elementar volume \(dV\).

Composite theorem for mass centre#

../_images/71.png
../_images/81.png

where,

  • \(r^*_i\) is the position vector locating the mass centre of \(S_i\), the \(i^{\text{th}}\) system of particles.

  • \(m_{s_{i}}\) is the mass of \(i^{\text{th}}\) system.

  • \(r^*\) is the mass centre of the composite system \(S\).

Example #1#

../_images/91.png

Given:

  • \(F\) and \(R\) are the bodies of mass density \(\rho\; kgm^{-2}\) and \(\sigma\;kgm^{-1}\) respectively.

  • \(P\) is a particle of mass \(m\).

Find:

  • Mass centre of the combined system.

Example #2#

\(F\) is split into two: \(F_1\) and \(F_2\).

\(m_{F_{1}}\) = \(\rho H_a\), is mass of \(F_1\).

\(m_{F_{2}}\) = \(\rho B_a\), is mass of \(F_2\).

Also,

\(m_{R}\) = \(\sigma L\), is mass of \(R\).

Then,

../_images/101.png

Similarly,

../_images/111.png

and

../_images/121.png

Inertia scalar#

For a particle \(P\) of mass \(m\), we can define a parameter called the inertia scalar. This is defined relative to an arbitrary point \(O\). There are two such inertia scalars:

../_images/131.png

1. Product of inertia#

Notation#

\(I^{P/O}_{ab}\) is the product of inertia of \(P\) along two lines through point \(O\) that are parallel to unit vectors \(\hat{n}_a\) and \(\hat{n}_b\).

../_images/141.png

8.5 can be extended for both systems of particles and continua.

Product of inertia of system particles#

../_images/151.png

Product of inertia of continua#

../_images/161.png
../_images/171.png

Warning

In all cases, \(I_{ab} = I_{ba}\) because the formula relies on the dot product of vectors.

2. Moment of inertia#

../_images/181.png

Notation#

\(I^{P/O}_{aa}\) is the moment of inertia of P about a line through point \(O\) which is parallel to the unit vector \(\hat{n}_a\).

../_images/191.png

8.8 can be extended to both systems of particles and continua.

Moment of inertia of system of particles#

../_images/201.png

Moment of inertia of system of particles#

../_images/211.png
../_images/221.png

Example#

../_images/231.png

  • \(P\) is a particle of mass \(m\).

  • \(\hat{n}_x,\;\hat{n}_y,\;\hat{n}_z\) are unit vectors that are mutually orthogonal.

  • \(\vec{r} = x\hat{n}_x + y\hat{n}_y + z\hat{n}_z\)

Find:

\[ I^{P/O}_{xx} \qquad I^{P/O}_{yy} \qquad I^{P/O}_{zz} \]

Solution

../_images/241.png

From inertia scalars to inertia matrix#

  • From the previous example, we now have some insight that we will be interested in computing the moments of inertia and products of inertia about a set of unit vectors that make up a reference frame.

  • For this discussion, we assume that the unit vectors are: \(\hat{n}_x,\;\hat{n}_y,\;\hat{n}_z\).

  • The inertia scalars can be used to define a square matrix called the inertia matrix.

Notation:

  • \(\left[I\right]^{S/O}\) is the inertia matrix of \(S\), a system of particles about the point \(O\).

  • The diagonal elements of this matrix are the moments of inertia.

  • The off-diagonal elements are the products of inertia.

  • So, the inertia matrix is represented as:

../_images/251.png

Warning

In all cases, \(I_{ab} = I_{ba}\) because the formula relies on the dot product of vectors.

  • Rigid body/ continua: The inertia scalars of a rigid body can also be arranged into an inertia matrix.