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 \[ (hf-bg)/(ab-h^2); \quad (gh-af)/(ab-h^2). \]
If \(Q=ax^2+2bx+c\), and \[ I_n = \int \frac{dx}{Q^{n+1}}, \] show by differentiating \((Ax+B)/Q^n\) (where \(A, B\) are adjustable constants), or otherwise, that \[ 2n(ac-b^2)I_n = \frac{ax+b}{Q^n} + (2n-1)aI_{n-1}. \] Obtain a similar formula of reduction for \[ J_n = \int \frac{x\,dx}{Q^{n+1}}. \] Evaluate \[ \int_0^1 \frac{dx}{(x^2-x+1)^3}. \]
From the parallelogram of forces show that, if two couples acting in a plane are in equilibrium, their moments are equal and opposite. Show, conversely, that two co-planar couples of equal and opposite moment are in equilibrium. A force acting in a plane has moments \(M_1, M_2, M_3\) about points whose coordinates referred to axes \(Ox, Oy\) in the plane are \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) respectively. Show that the equation of the line of action of the force is \[ x \begin{vmatrix} 1 & y_1 & M_1 \\ 1 & y_2 & M_2 \\ 1 & y_3 & M_3 \end{vmatrix} + y \begin{vmatrix} x_1 & 1 & M_1 \\ x_2 & 1 & M_2 \\ x_3 & 1 & M_3 \end{vmatrix} = \begin{vmatrix} x_1 & y_1 & M_1 \\ x_2 & y_2 & M_2 \\ x_3 & y_3 & M_3 \end{vmatrix}. \]
A light inelastic string, of length \(2l\), is fixed at its upper end; it carries a particle of mass \(m\) at its mid-point and a particle of mass \(M\) at its lower end. The particles move in a vertical plane so that the upper and lower portions of the string make angles \(\theta\) and \(\phi\) respectively with the vertical, and on the same side of it. If the angular displacements \(\theta, \phi\) are small, write down the equations of motion of \(m\) and \(M\), neglecting quantities of order higher than the first. Show that solutions of these equations can be obtained by assuming that \(\phi=k\theta\), where \(k\) is a constant. In particular, describe the corresponding motions when \(m=3M\).