A uniform triangular lamina has mass \(M\) and sides \(a\), \(b\) and \(c\). Find its moment of inertia about the axis through its mass centre and perpendicular to its plane. [Hint: divide the triangle into four equal similar ones.]
A particle is projected in a given vertical plane from an origin \(O\), with velocity \((2gh)^{1/2}\). It passes through the point \((x, y)\) at time \(t\) after projection, the axes being horizontal and vertically upwards. Show that \[p^2 - 2(2h - y)p + r^2 = 0,\] where \(p = \frac{1}{2}gt^2\) and \(r^2 = x^2 + y^2\). Show that the points of the plane which are accessible from \(O\) by projection with the given velocity lie on or under the parabola having \(O\) as focus, and its vertex a distance \(h\) vertically above \(O\), and that the time taken to reach a point on this parabola is \((2v/g)^{1/2}\).
In three-dimensional Euclidean space, \(\mathbf{u}\) is a fixed vector of unit length, and \(\mathbf{r}\) is a given vector. Using the notation of scalar and vector products, show how to write the sum of a part parallel to \(\mathbf{u}\) and a part perpendicular to \(\mathbf{u}\). Hence, or otherwise, show that if the plane containing \(\mathbf{r}\) and \(\mathbf{u}\) is rotated through an angle \(\phi\) measured in the clockwise sense relative to the direction of \(\mathbf{u}\), and \(\mathbf{r}\) is thereby transported to a new position \(\mathbf{r}'\), then \[\mathbf{r}' = \mathbf{r}\cos\phi + \mathbf{u}(\mathbf{r} \cdot \mathbf{u})(1 - \cos\phi) + (\mathbf{u} \times \mathbf{r})\sin\phi.\]
A spacecraft may be regarded as a solid body which is convex (i.e. no straight line meets its surface more than twice), and its total surface area is \(A\). It is required to measure a certain type of radiation. If the radiation has a certain strength, and is unidirectional (i.e. incident on the spacecraft in the form of parallel rays), the response of the detector on board is given by \(Sk\), where \(S\) is the cross-sectional area presented by the spacecraft in that direction and \(k\) is a constant. If now the spacecraft is subjected to radiation of the same total strength but isotropic (i.e. scattered equally in all directions), show that the response is \(\frac{1}{4}Ak\), whatever the shape of the spacecraft.