A unit mass at \(P\) moves in a horizontal straight line \(Ox\), and is subject to a force \(n^2x\) directed towards \(O\) where \(OP=x\), and to a resisting force which acts only if \(\dot{x}\) is positive, and has constant value \(na\) where \(a\) is positive. The mass is released from rest when \(x\) is negative and \(|x|=b\). Indicate briefly the nature of the subsequent motion and show that exactly \(r\) half-swings will be made before the mass comes finally to rest where \[ ra < (b^2+a^2)^{\frac{1}{2}} < (r+1)a. \]
In an exhibition of motor cycling on a ``wall of death'' the cyclist describes a horizontal circle with constant velocity with the wheels in contact with the walls of a rough surface of revolution with a vertical axis. If \(\mu\) is the coefficient of friction between the wheels and the wall, \(\psi\) the inclination to the horizontal of the tangent to the meridian section of the surface of revolution at the point of contact, and \(\rho\) the radius of curvature of the path of the joint centre of mass of the cyclist and machine, show that to avoid skidding the velocity \(v\) of this centre must lie between two values \(v_1\) and \(v_2\) if \(\psi<45^\circ\) and \(\mu<\tan\psi\). Find these values in terms of \(\rho, \psi\) and \(\mu\). Show that if \(\psi\) is unrestricted, then if \(\mu>\tan\psi\), \(v\) has no lower limit; and if \(\mu>\cot\psi\), \(v\) has no upper limit. (The distance between the wheels may be neglected in comparison with \(\rho\)).
A small smooth sphere of mass \(m\) hangs at rest from a point \(O\) by a light inelastic string of length \(a\). Another small sphere of mass \(M\) is allowed to slide from rest at a point of a smooth rigid tube, bent in the form of a semicircle centre \(O\) and radius \(a\) with diameter vertical, and to strike the first sphere with direct impact. Prove that, if in the subsequent motion the suspended sphere reaches the point at a height \(a\) vertically above \(O\), then \[ \frac{m}{M} \le \frac{4-\sqrt{5}}{\sqrt{5}}, \] and the coefficient of restitution must exceed the value \(\frac{1}{2}(\sqrt{5}-2)\).
A non-uniform sphere of radius \(a\) whose centre of mass is at the geometric centre and whose radius of gyration about any diameter is \(k\), is released from rest on a rough plane inclined at an angle \(\alpha\) to the horizontal. Prove that the sphere will roll down the plane if the coefficient of friction exceeds a certain critical value, but will skid if the coefficient is less than this value, and find this critical value. Show that the acceleration of the centre of the sphere is less in the first case than in the second case, while the angular acceleration is greater in the first case. Verify that the difference in the angular acceleration is \(a/k^2\) times the difference in the linear acceleration.
By factorizing the determinant, or otherwise, show that \[ \begin{vmatrix} x & y & z & u \\ u & x & y & z \\ z & u & x & y \\ y & z & u & x \end{vmatrix} = (x^2+z^2-2yu)^2 - (u^2+y^2-2zx)^2. \] Express \[ \{(x^2+z^2 - 2yu)^2 - (u^2 + y^2 - 2zx)^2\} \{(X^2 + Z^2 - 2YU)^2 - (U^2 + Y^2 - 2ZX)^2\} \] in the form \((A^2+C^2-2BD)^2 - (D^2+B^2-2CA)^2\), giving explicit expressions for \(A, B, C, D\) in terms of \(x, y, z, u\) and \(X, Y, Z, U\).
Let \(N(n)\) denote, for any given integer \(n\) (positive, zero, or negative) the number of solutions of the equation \[ x+2y+3z=n \] in non-negative integers \(x, y, z\) (so that \(N(n)=0\) for \(n<0\), \(N(0)=1\), \(N(1)=1\), \(N(2)=2\), etc.). By considering the coefficient of \(t^n\) in the expansion of \[ \frac{1-t^6}{(1-t)(1-t^2)(1-t^3)} \] in ascending powers of \(t\), or otherwise, prove that \[ N(n) - N(n-6) = n \quad (n>0), \] and write down the corresponding formula for \(n=0\). Defining the integers \(q, r\) by \[ n = 6q+r \quad (0 \le r < 6), \] obtain an expression (or expressions) for \(N(n)\) (\(n \ge 0\)) in terms of \(n\) and \(r\). Show that, for every \(n \ge 0\), \(N(n)\) is the integer nearest to \(\frac{1}{12}(n+3)^2\).
Prove the following inequalities:
If \(z = \frac{y}{x} f(x+y)\) and subscripts denote partial differentiations, show that \begin{align*} xz_x + yz_y &= \frac{y}{x}(x+y)f'(x+y); \\ x^2z_{xx} + 2xyz_{xy} + y^2z_{yy} &= \frac{y}{x}(x+y)^2f''(x+y), \end{align*} in which \(f'(t)\) stands for \(df(t)/dt\), and so on. Find also the value of \[ x^3z_{xxx} + 3x^2yz_{xxy} + 3xy^2z_{xyy} + y^3z_{yyy}. \]
A conic \(K\) touches four straight lines \(a, b, c, d\) at \(A, B, C, D,\) respectively. Prove that there is a conic \(S\) through the six points \(A, B, C, D, ab, cd\) (where \(ab\) is the intersection of \(a\) and \(b\)), and a conic \(\Sigma\) touching the six lines \(a, b, c, d, AB, CD\). Show that the tangents to \(K\) at its four points of intersection with \(\Sigma\) touch \(S\).
Show that the condition that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] should represent two straight lines is \[ \Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0. \] Show further that necessary conditions that these lines should be real are \(h^2 \ge ab\); \(f^2 \ge bc\); \(g^2 \ge ca\). If \(\Delta=0\), prove that the point of intersection of the lines is \[ (hf-bg)/(ab-h^2); \quad (gh-af)/(ab-h^2). \]