Composition of Quantities

Excerpts

• Composition of Quantities

  The equation states that the mass $m$ at the extremity of the vector $A$ is equivalent to the equal mass at the extremity of $R$, together with the said mass-vector applied at the extremity of $R$.

• Composition of Quantities

  This equation asserts that a force $F$ applied at the extremity of $A$ is equivalent to an equal force applied at the extremity of $R$ together with a couple whose magnitude and direction are given by the vector product of the radius-vector from the extremity of $R$ to the extremity of $A$ and the force.

• Composition of Quantities

  The product $m(A - R)$ is what Clerk Maxwell called a mass-vector, and means the directed moment of $m$ with respect to the extremity of $R$.

• Composition of Quantities

  The term (2) means the projection of $R$ upon that line.

• Composition of Quantities

  This is the same straight line as before, only no relation is now imposed on the directions of $\sum F$ and $\sum \mathrm{V}AF$; hence there always is a central axis.

Equations

• Composition of Quantities

  \sum m_A = \sum m_R + \sum\Bigl\{m(A - R)\Bigr\}

  For any number of masses, the total mass at A equals the total mass at R plus the sum of the mass-vectors about R.

• Composition of Quantities

  R = \frac{\sum mA}{\sum m},\quad\text{or}\quad R \sum m = \sum mA

  The resultant moment of the masses vanishes when R is the point at the weighted mean position of the masses, which is the centre of mass.

• Composition of Quantities

  x &= \frac{\sum (ma)}{\sum m}

  The x-coordinate of the centre of mass is the mass-weighted mean of the x-coordinates of the masses.

• Composition of Quantities

  y = \frac{\sum (mb)}{\sum m}

  The y-coordinate of the centre of mass is the mass-weighted mean of the y-coordinates of the masses.

• Composition of Quantities

  z = \frac{\sum (mc)}{\sum m}

  The z-coordinate of the centre of mass is the mass-weighted mean of the z-coordinates of the masses.

• Composition of Quantities

  A &= a_1j + b_1j+c_1k;

  The radius-vector A is written in components a, b, c along the coordinate axes.

• Composition of Quantities

  F_A &= F_R + \mathrm{V}(A-R)F

  A force F applied at A is equivalent to an equal force at R together with a couple given by the vector product of the vector from R to A and the force.

• Composition of Quantities

  \sum \left(F_A\right) &= \sum \left(F_R\right) + \sum \mathrm{V}\left(A - R\right)F

  For a system of forces at different points, the forces at A equal the forces at R plus the sum of the couples about R.

• Composition of Quantities

  \mathrm{V} R \sum F &= \sum \mathrm{V} A F

  There is no resultant couple when the vector product of R with the resultant force equals the sum of the vector products of each point with its force.

• Composition of Quantities

  R &= \frac{1}{\sum F}\sum \left(\mathrm{V}AF \right) \tag{1}

  The vector term (1) is perpendicular to the resultant force and fixes the line along which R must lie for the resultant couple to vanish.

• Composition of Quantities

  \mathrm{V}\left\{\sum \mathrm{V}AF - \mathrm{V}R\sum F\right\} \sum F = 0

  On the central axis the resultant force and the resultant couple have the same direction.

Problems