10273 problems found
A heavy elastic string, of length \(l\), would have its length doubled by a pull equal to its own weight. Find its length when hanging vertically from one end. One end of this string is fixed to a point on a rough plane inclined at an angle \(\alpha\) to the horizontal, and the string is placed down a line of greatest slope. Shew that the minimum increase in the length of the string is \(l\frac{\sin(\alpha-\lambda)}{\cos\lambda}\), where \(\lambda\) is the angle of friction (\(\lambda < \alpha\)).
\(OP, OQ\) are conjugate semidiameters of \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\). The circles of curvature at \(P, Q\) meet the ellipse again at \(P', Q'\) respectively. Prove that the locus of the point of intersection of \(PP', QQ'\) is \[ \left(\frac{x^2}{a^2} + \frac{y^2}{b^2}\right)^3 = 8\frac{x^2y^2}{a^2b^2}. \]
Shew that, if \(n=3\), \(a+b+c\) is a factor of \[ \begin{vmatrix} a^n & b^n & c^n \\ a & b & c \\ 1 & 1 & 1 \end{vmatrix}, \] and that, if \(n\) is greater than 3, \(a+b+c\) is not a factor.
Shew that a system of particles has one and only one centre of mass. Find the centres of mass of the following solids of uniform density, (1) a tetrahedron; (2) a pyramid with a plane base. A solid is bounded by five faces, namely, a parallelogram \(ABCD\), a trapezium \(ABFE\) in which \(AB, EF\) are parallel, a trapezium \(CDEF\), a triangle \(AED\) and a triangle \(BFC\). Prove that the centre of mass of the solid divides the join of the middle point of \(EF\) and the centre of the parallelogram \(ABCD\) in the ratio \(3AB+EF : AB+EF\).
Shew that, by plotting a curve connecting the reciprocal of the acceleration of a body with its velocity, it is possible to estimate the time required for a given change of velocity. The acceleration of a tramcar starting from rest decreases by an amount proportional to the increase of speed, from 1.5 f.s.s. at starting to 0.5 f.s.s. when the speed is 5 m.p.h. Find the time taken to reach 5 m.p.h. from rest.
\(n\) quantities are given. \(s_r\) denotes the sum of the products of all combinations of the quantities \(r\) at a time. Prove that the sum of the products of all combinations of the squares of the quantities \(m\) at a time is given by \[ s_m^2 - 2s_{m-1}s_{m+1} + 2s_{m-2}s_{m+2} + \dots + (-)^{m-1}2s_1s_{2m-1} + (-)^m 2s_{2m} \] where we suppose \(2m \le n\).
An approximate value for the angle \(\phi\), measured in radians, is \(\displaystyle\frac{3 \sin\phi}{2 + \cos\phi}\), provided \(\phi\) is less than \(\frac{1}{2}\pi\). Establish this result when \(\phi\) is small, and shew that the error is approximately \(\displaystyle\frac{\phi^5}{180}\). Hence, express approximately the acute angles of a right-angled triangle in terms of the sides, and deduce that \[ \frac{6(a+b)c + 3c^2}{ab + 2(a+b)c + 4c^2} \] is nearly equal to \(\frac{1}{2}\pi\) for all positive values of \(a\) and \(b\), provided \(c^2 = a^2+b^2\).
Explain what is meant by the shearing stress and bending moment in a beam, and obtain the relations between them and the load. Discuss the characteristics of the distributions of shearing stress and bending moment along a beam which is loaded at various points and supported in a horizontal position at two points when the beam is (a) light, (b) uniformly heavy. In case (a) shew that the funicular polygon for the loads and reactions is also a bending moment diagram such that an intercept by the polygon on a line parallel to the loads is a measure of the bending moment at the corresponding point of the beam.
A particle is projected vertically upwards with velocity \(V\), and the resistance of the air produces a retardation \(kv^2\), where \(v\) is the velocity. Shew that the velocity \(V'\) with which the particle will return to the point of projection is given by \[ \frac{1}{V'^2} = \frac{1}{V^2} + \frac{k}{g}. \]
Prove the formulae