A homogeneous sphere impinges obliquely upon a horizontal plane which is so rough that the sphere rolls rather than slides on it during contact. If before impact the sphere has no spin and the deformation of both the sphere and the plane is negligible during impact, show that the horizontal component of the sphere's velocity after impact is \(\frac{5}{7}\) of the horizontal component before impact. [It may be assumed that the moment of inertia of a homogeneous sphere of mass \(m\) and radius \(a\) about any diameter is \(\frac{2}{5}ma^2\).] A bowler bowls a cricket ball so that it leaves his hand horizontally at a speed of 54 m.p.h., from a height of 8 ft., with no spin. If the pitch is so rough that the ball cannot slide upon it, find, correct to two significant figures, the coefficient of restitution for the impact of the ball on the pitch if the ball hits the wicket when it has risen through a height of 1 ft. 6 in. after bouncing. [Air friction should be neglected and the ball should be regarded as a homogeneous sphere whose radius is negligible in comparison with 8 ft. The length of a cricket pitch is 22 yds.]
A smooth thin wire of mass \(M\) has the form of a circle of radius \(a\). It is constrained so that a certain diameter is vertical, but it can spin freely about this diameter. A small bead of mass \(m\) is free to slide on the wire. Initially the wire spins with angular velocity \(\Omega\) and the bead is at rest at the lowest point of the wire. Show that, if the bead is displaced slightly from its initial position, it will perform simple harmonic oscillations of period $$2\pi \left(\frac{g}{a} - \Omega^2\right)^{-\frac{1}{2}},$$ provided that \(\Omega^2 < g/a\).
Show that the distance of the point \(\mathbf{a}\) from the plane $$\mathbf{r} \cdot \mathbf{n} = p,$$ where \(\mathbf{n}\) is a unit vector, is $$|\mathbf{a} \cdot \mathbf{n} - p|.$$ A circle \(S\) is defined by the intersection of the surfaces $$\mathbf{r} \cdot \mathbf{n} = p, \quad (\mathbf{r} - \mathbf{c})^2 = R^2.$$ Show that, if \(\mathbf{c} \cdot \mathbf{n} = p\), the distance between the point \(\mathbf{a}\) and the closest point of \(S\) is $$\{(\mathbf{a} - \mathbf{c})^2 + R^2 - 2R[(\mathbf{a} - \mathbf{c})^2 - (\mathbf{a} \cdot \mathbf{n} - p)^2]^{\frac{1}{2}}\}^{\frac{1}{2}}.$$
A substance \(A\) changes into a substance \(B\) at a rate of \(\alpha\) times the amount of \(A\) instantaneously present; \(B\) changes back into \(A\) at a rate of \(\beta\) times the amount of \(B\) present, and into \(C\) at a rate of \(\gamma\) times the amount of \(B\) present. If initially there is an amount \(X\) of \(A\) and no \(B\) or \(C\), show that the amount of \(C\) after a time \(t\) is $$X\left[1 - \frac{me^{mt} - ne^{nt}}{m-n}\right],$$ where \(m\) and \(n\) are the roots of the equation $$(z + \alpha)(z + \beta + \gamma) - \alpha\beta = 0.$$