Explain how Newton's laws of motion enable the concepts of ``mass'' and ``force'' to be defined in terms of observable quantities. Discuss the status of the hypothesis of gravitation in relation to Newton's laws, with particular reference to Newton's choice of an inertial frame of reference.
A die marked with the numbers 1, \dots, 6 is thrown \(r\) times and the \(r\) numbers obtained are added. If the numbers 1, \dots, 6 are equally likely to be obtained on each throw show, by considering the coefficient of \(x^n\) in \((x+x^2+\dots+x^6)^r\), that the probability that the sum of the \(r\) numbers should be \(n\) is \[ \frac{1}{6^r}\left[ \frac{r(r+1)\dots(n-1)}{(n-r)!} - r\frac{r(r+1)\dots(n-7)}{(n-r-6)!} + \frac{r(r-1)}{2!}\frac{r(r+1)\dots(n-13)}{(n-r-12)!} - \dots \right], \] and hence find the probability that the total after four throws should be 14.
The three complex numbers \(z_1, z_2, z_3\) are represented in the Argand diagram by the vertices of a triangle \(Z_1Z_2Z_3\) taken in counterclockwise order. On the sides of \(Z_1Z_2Z_3\) are constructed isosceles triangles \(Z_2Z_3W_1, Z_3Z_1W_2, Z_1Z_2W_3\), lying outside \(Z_1Z_2Z_3\). The angles at \(W_1, W_2, W_3\) all equal \(2\pi/3\). Find the complex numbers represented by \(W_1, W_2, W_3\) and prove that the triangle \(W_1W_2W_3\) is equilateral.
Let \[ \rho = \cos\frac{2\pi}{m} + i\sin\frac{2\pi}{m}, \] where \(m\) is a positive integer. For any integer \(r\) put \[ p_m(r) = \frac{\rho^r}{1-\rho} + \frac{\rho^{2r}}{1-\rho^2} + \dots + \frac{\rho^{(m-1)r}}{1-\rho^{m-1}}. \] By considering the differences \[ p_m(r+1) - p_m(r) \] and the sum \[ p_m(0)+p_m(1)+\dots+p_m(m-1), \] or otherwise, evaluate the \(p_m(r)\) for all \(m\) and \(r\). Show in particular that \[ p_m(0) = \frac{1}{2}(m-1). \]
Two lines \(h\) and \(k\) cut at right angles, \(T\) is a point of their plane, and \(A\) is a fixed point of \(h\). The circle on \(AT\) as diameter meets \(k\) in \(L\) and \(M\), and \(TL, TM\) meet \(h\) in \(L', M'\) respectively. Points \(L'', M''\) are taken on \(L'L, M'M\) respectively, such that \(L'L=LL''\) and \(M'M=MM''\). Show that the triangles \(AL''T\) and \(ATM''\) are similar. If \(T\) is allowed to vary, what is the locus of \(L''\) (or \(M''\))?
The normal to the rectangular hyperbola \(S\) at the point \(P\) cuts \(S\) again in \(N\); the diameter through \(P\) cuts \(S\) again in \(P'\). Prove that \(PP'\) is perpendicular to \(P'N\). The pole of \(PN\) with respect to \(S\) is the point \(Q\); prove that \(QP=QP'\).
A rectangular picture frame hangs from a smooth peg by a string of length \(2a\) whose ends are attached to two points on the upper edge at distances \(c\) from its middle point. Prove that if the depth of the frame exceeds \(2c^2(a^2-c^2)^{-1/2}\) there is no position of equilibrium except that in which the picture frame hangs symmetrically.
A uniform flexible chain of length \(6l\) hangs in equilibrium over two small smooth pegs at the same horizontal level with a length \(2l\) of chain between the pegs. Prove that the distance between the pegs must be \((\frac{4}{3}\log 3)l\), and find the depth of the free ends of the chain below its mid-point.
A particle hangs in equilibrium from the ceiling of a stationary lift, to which it is attached by an elastic string (of natural length \(l\)) extended to length \(l+a\). The lift descends, moving for time \(T\) with constant acceleration \(f\) and subsequently with constant velocity. Prove that, if \(f < g\), the string never becomes slack, and show that the amplitudes of the oscillations before and after time \(T\) are respectively \(af/g\) and \(2af|\sin\frac{1}{2}nT|/g\), where \(n^2=g/a\).
A particle is projected horizontally with speed \(\sqrt{(\lambda ag)}\), where \(0<\lambda<1\), from the highest point of a fixed smooth sphere of radius \(a\). Find the velocity of the particle at the instant it leaves the sphere. After leaving the sphere the particle describes a parabola. Find the depth of the vertex of this parabola below the point of projection.