10273 problems found
A particle hangs from a light inextensible string of length \(r\) attached at its upper end to a point on a vertical wall. It is projected with velocity \(2\sqrt{rg}\) perpendicular to the wall. Find the point at which the particle will subsequently hit the wall.
A uniform disc of radius \(r\) and mass \(M\) is freely pivoted at a point on its circumference and hangs in a vertical plane. It is struck by a horizontal impulse \(F\) which acts in the plane of the disc, and is distant \(h\) below the pivot. Find the impulsive force at the pivot, and prove that if \(h=r\) the energy imparted to the disc is \(F^2/3M\).
A light uniform rod of length \(2l\) is freely suspended from one end \(A\) and carries a concentrated mass \(m\) at the other end \(B\). A light spring, producing a force \(\frac{16mg}{l}\) per unit extension or compression, is attached to the rod at its mid-point and lies horizontally so that the rod is vertical when in equilibrium. Find the amplitude and period of small oscillations of the end \(B\) (i) when it is given a small displacement \(a\) in a direction perpendicular to the spring and then set free; and (ii) when it is projected with a small velocity \(v\) in a direction parallel to the spring. Sketch the projection on a horizontal plane of the path of \(B\) when these two oscillations are superposed, assuming them to begin simultaneously.
Prove one of the following theorems and deduce the other from it.
Describe and prove the funicular polygon method of finding graphically the line of action of the resultant of a set of co-planar non-concurrent forces. Consider, in particular, the case of parallel forces.
Prove that the inverse of a circle (with respect to a coplanar circle) is a circle or a straight line. Circles \(S\) are drawn to touch two given coplanar intersecting circles \(C_1, C_2\). Show that, if two of the circles \(S\) touch, their point of contact (supposed not on \(C_1\) or \(C_2\)) lies in general on one of two fixed circles.
Sum the series \[ \sum_{r=1}^n \frac{1}{r(r+1)(r+2)}, \quad \sum_{r=1}^\infty \frac{r}{2^r}, \quad \sum_{r=1}^\infty \frac{1.3.5 \dots (2r-1)}{3.6.9 \dots 3r}. \]
If three conics have double contact in pairs, prove that the extremities of each chord of contact form a harmonic range with the points where this chord meets the other two chords of contact. Prove, conversely, that if pairs of points \((A, A')\), \((B, B')\), \((C, C')\) are taken in a plane so that \(A, A'\) are harmonically conjugate with respect to the points where \(AA'\) meets \(BB'\) and \(CC'\), and similarly for the other pairs, then a set of three conics can in general be found having double contact in pairs at the ends of the segments \(AA', BB', CC'\).
A hollow circular cylinder, of weight \(W'\), is made of uniform thin sheet material and is open at both ends, whose planes are perpendicular to the axis; the cylinder rests with its axis vertical on a smooth horizontal plane. Three equal smooth uniform solid spheres, each of weight \(W\), are placed inside the cylinder, and their centres are \(A, B, C\). The lowest sphere \(A\) lies on the table; the sphere \(B\) lies above it, with \(AB\) inclined at an angle \(\alpha\) (\(< \pi/3\)) to the vertical. The third sphere is uppermost, and \(CA\) is vertical. If the reactions of the spheres on the cylinder do not cause it to overturn, show that \(2W \sin\alpha < W'(1 + \sin\alpha)\).
Prove that the reciprocal of a conic with respect to a focus is a circle. A variable chord of a fixed conic subtends a right angle at a given point \(O\) in the plane of the conic but not on the curve. Show that the chord envelops a conic having a focus at \(O\). In what circumstances is the envelope (i) a parabola, (ii) a circle?