Week 10 Exercises

10.3 Sheet 19 Centroids of area

Exercise 1

Determine the positionns of the centroids of the uniform laminae represented by the following curves and indicated boundaries.

  1. \(y=x^2\), the \(x\)-axis and \(x=0, x=2\);
  2. \(y=2x^3\), the \(x\)-axis and \(x=1, x=2\);
  3. The portion of \(y=2x-x^2\) above the \(x\)-axis.

Exercise 2

Determine the position of the centroid of the figure bounded by \(y = e^{2x}\), the \(x-\)axis, the \(y-\)axis and the ordinate at \(x = 2\).

Exercise 3

Determine the position of the centroid of the figure bounded by the curve \(y = 5\sin(2x)\), the \(x-\)axis and the ordinates at \(x = 0\) and \(x = \pi/6\).

Exercise 4

A curve is given parametrically as \(x = at^2, y = 2at\) where \(t\) is a parameter and \(a\) is a constant. Determine the position of the centroid about the for figure bounded by the values \(t = 1\) and \(t = 2\).

[Solutions: 1. (i) \(\overline{x}=1.5\), \(\overline{y}=1.2\), (ii) \(\overline{x}=1.65\), \(\overline{y}=4.8\), (iii) \(\overline{x}=1\), \(\overline{y}=0.4\); 2. \(\overline{x}=1.5\), \(\overline{y}=13.6\); 3. \(\overline{x}=0.3\), \(\overline{y}=1.5\); 4. \(\overline{x}=2.66\), \(\overline{y}=1.61\);]

10.4 Sheet 21 Vectors I

Exercise 1

Two vectors \(\underline A\) and \(\underline B\) are at right angles. Find

  1. the magnitude of the vector \(\underline A+\underline B\) if \(|\underline A| = 1\) and \(|\underline B| = 2\).
  2. the magnitude of the vector \(\underline A+\underline B\) if \(|\underline A|=20\) and \(|\underline B| = 21\).

Exercise 2

Determine the resultant of the vectors \(\underline A, \underline B, \underline C\) and \(\underline D\) given that \(\underline A = 20\) km \(3o^\circ\) S of E, \(\underline B = 50\) km due W, \(\underline C = 40\) km NE and \(\underline D=30\) km SW.

Exercise 3

An object \(P\) is acted on by three forces as shown on Figure 10.7. All forces lie in the same plane (are co-planar). Determine the required force and its direction so as to prevent \(P\) from moving.
resultant force

Figure 10.7: resultant force

Exercise 4

For the triangle on Figure 10.8, prove by vectors that the line \(DE\), where \(D\) and \(E\) are the mid-points of \(AB\) and \(AC\) respectively, is half the length of the side \(BC\).
triangles

Figure 10.8: triangles

Exercise 5

The quadrilateral \(ABCD\) shown on Figure 10.9 has the mid-points of the diagonals as \(P\) and \(Q\). Show that \(\underline{AB}+\underline{AD}+\underline{CB}+\underline{CD} = 4\underline{PQ}\)
quadrilateral

Figure 10.9: quadrilateral

[Solutions: 1. (i) \(\sqrt{5}\), (ii) \(29\); 2. \([25.7 \angle -173^{\circ}]\); 3. \(-323\underline{i}\,N\);]