10273 problems found
If in a triangle \(ABC\) the side \(a\) is increased by a small quantity \(x\) while the other two sides are unaltered, shew that the radius of the circumscribing circle will be increased by \[ \frac{1}{2}x \operatorname{cosec} A \cot B \cot C. \]
Two particles of equal mass joined by a light inextensible string of length \(\pi r/2\) rest in (unstable) equilibrium on the outer surface of a smooth circular cylinder of radius \(r\) whose axis is horizontal. If the equilibrium is slightly disturbed and the particles begin to move in a vertical plane, prove that the first particle to leave the surface of the cylinder does so when the perpendicular drawn from it to the axis of the cylinder is inclined at about 13\(^\circ\) to the horizontal.
A smooth rod makes an angle \(\alpha\) with the horizontal. A ring of mass \(m\) can slide along the rod and is attached by a fine string passing over a smooth pulley to a particle of mass \(m'\). The pulley is below the rod and the string and rod are in a vertical plane. The ring begins to move along the rod when the string between it and the pulley is horizontal and of length \(a\). Shew that, if \(m \sin\alpha > m'\), the ring will not come to rest however long the string may be, and find the distance that the ring moves before coming to rest when \(m \sin\alpha < m'\).
A quadrilateral whose sides are of lengths \(a,b,c,d\) is inscribed in a circle. Prove that the lengths of the diagonals are \[ \{(ac+bd)(ad+bc)/(ab+cd)\}^{\frac{1}{2}}, \] and \[ \{(ac+bd)(ab+cd)/(ad+bc)\}^{\frac{1}{2}}, \] and that the product of the segments of a diagonal is \[ abcd(ac+bd)/\{(ab+cd)(ad+bc)\}. \]
A uniform circular wire of mass \(m\) and radius \(r\) can rotate freely about a fixed vertical diameter, and a small ring of mass \(m\) can move freely along the wire. The wire is started rotating with angular velocity \(\omega\), at an instant when the ring is at one end of a horizontal diameter and is at rest with respect to the wire: if the system is left to itself in the subsequent motion, shew that, provided \(\omega^2 < 2g/3r\), the ring reaches the lowest point of the wire with velocity \[ (2gr-3\omega^2r^2)^{\frac{1}{2}}. \] Describe the motion of the ring if \(\omega^2>2g/3r\).
A particle of mass \(m\) is attached by a light spring to a fixed point on a smooth horizontal board of mass \(M\) which can slide in a prescribed direction on a horizontal plane. The spring is capable of longitudinal extension and compression in the prescribed direction. Shew that when motion takes place the number of oscillations per second is \(\sqrt{}(1+m/M)\) of what it would be if the board were fixed.
(a) Show that if two curves are polar reciprocals in the circle \(r=a\) their radii of curvature at corresponding points are connected by the relation \(\rho_1 \rho_2 = \frac{r_1^3 r_2^3}{a^4}\), where \(r_1, r_2\) are the distances from the pole to the corresponding points, and \(p_1, p_2\) the radii of curvature. (b) If two curves are inverses in \(r=a\), show that \(r\frac{d^2p}{dr^2} - \frac{dp}{dr}\) has the same value at corresponding points on each of them.
Prove that four normals can be drawn to an ellipse from a given point. Normals are drawn through the centre of curvature \(P\) of a point on the ellipse whose eccentric angle is \(\alpha\). Shew that the eccentric angles of the other points the normals at which pass through \(P\) are the values of \(\theta\) given by \[ \frac{\sin\theta}{\sin\alpha} + \frac{\cos\theta}{\cos\alpha} + 1 = 0. \]
A light inextensible string \(BC\) joins the ends of two uniform rods \(AB\) and \(CD\) which are of the same length. The system is placed on a smooth horizontal table so that \(AB, BC\) and \(CD\) form three sides of a rectangle, and an impulse \(I\) in the direction \(AD\) is then applied to the rod \(AB\) at \(A\). Prove that the initial velocity of \(D\) is \(I/2m\), \(m\) being the mass of each rod.