A Venetian blind is 7 feet long when fully stretched out, and 1 foot long when completely drawn up. There are 30 movable strips, and each weighs one pound. Find the work done in raising the blind against gravity.
A rectangular picture frame hangs from a smooth peg by a string of length \(2a\) whose ends are attached to points on the upper edge at distances \(c\) from the middle point. Shew that if the depth of the picture exceeds \(\frac{2c^2}{\sqrt{a^2-c^2}}\), the symmetrical position is the only position of equilibrium. \par If the depth \(2d\) of the picture is less than the critical value given above, find the inclination of the picture to the horizontal in the other equilibrium position.
Forces \(X, Y, Z\) act along the sides \(BC, CA, AB\) of a triangle \(ABC\) (supposed not equilateral), and are such that the resultant is a force of magnitude \(P\) in the line joining the circumcentre \(O\) to the incentre \(I\) of the triangle \(ABC\). Prove that \[ \frac{X}{\cos B - \cos C} = \frac{Y}{\cos C - \cos A} = \frac{Z}{\cos A - \cos B} = \frac{R}{4OI\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}}P, \] where \(R\) is the circum-radius.
A circular cylinder of radius \(a\) and weight \(W\) rests with its axis horizontal in a V-shaped groove whose sides are inclined at equal angles \(\alpha\) to the horizontal, and a gradually increasing couple is applied to it in a plane perpendicular to its axis. If \(\alpha<\lambda\), where \(\lambda\) is the angle of friction, prove that the cylinder will begin to roll up one side of the groove when the couple exceeds \(Wa\sin\alpha\). If \(\alpha>\lambda\), prove that the cylinder will begin to rotate about its axis when the couple exceeds \[ \frac{1}{2}Wa\sec\alpha\sin 2\lambda. \]
An aeroplane has a speed \(u\), and a range of action \(R\) (out and home) in calm weather. If there is a north wind of velocity \(v\), prove that the range of action in a direction making an angle \(\phi\) with the north-south line is
\[ R \frac{u^2-v^2}{u\sqrt{u^2-v^2\sin^2\phi}}, \]
if \(v
A train of weight \(W\) is travelling with velocity \(v\) when the brakes are applied. The braking force increases at a constant rate from zero to a maximum \(\frac{W}{n}\) which is reached in a time \(T\), after which it remains constant until the train stops. Find the distance \(d\) travelled by the train before it comes to rest, and the time taken to stop the train. \par Sketch the graph of the variation of \(d\) with \(v\).
Two particles of masses \(M\) and \(m\) (\(M>m\)) are placed on the two smooth faces of a light wedge which rests on a smooth horizontal plane. The faces of the wedge are inclined to the horizontal at angles \(\alpha\) and \(\beta\) respectively. Initially the system is at rest. Shew that the smaller particle will move up the plane on which it is placed if \[ \tan\beta < \frac{M\sin\alpha\cos\alpha}{m+M\sin^2\alpha}. \]
Equal particles of mass \(m\) are attached to the ends of a light string \(AB\) which passes through a small smooth fixed ring \(O\), and rest on a smooth horizontal plane. The system is at rest with \(OA\) and \(OB\) taut, and \(OA=OB\). An impulse \(P\) is applied to the particle at \(A\) in a direction making an angle of \(60^\circ\) with the direction \(OA\). Find the initial motion of the particles, and shew that when the particle \(B\) reaches the ring its velocity is \(\frac{\sqrt{22}P}{8m}\).
Two particles \(A\) and \(B\) of masses \(m_1\) and \(m_2\) respectively are connected by a light spring. Prove that if the only force acting on each particle is due to the tension or thrust in the spring, the centre of mass describes a straight line with constant velocity. \par Shew further that the kinetic energy of the system is \[ \frac{1}{2}(m_1+m_2)V^2 + \frac{1}{2}\frac{m_1m_2}{m_1+m_2}v^2, \] where \(V\) is the velocity of the centre of mass, and \(v\) is the velocity of one particle relative to the other. \par The system rests on a smooth horizontal plane, and at time \(t=0\) the spring is in its natural position, the particle \(A\) is at rest, and \(B\) moves with velocity \(u\) in the line of the spring, away from \(A\). If \(m_1=m_2\), and the tension in the spring when its length is increased by unit distance is \(\frac{1}{2}m_1n^2\), shew that at time \(t\) the displacement of \(B\) is \[ \frac{1}{2}u\left(t+\frac{1}{n}\sin nt\right). \]
If \(p\) is a positive integer, shew that the number of distinct ways in which four positive (non-zero) integers may be chosen such that two and only two of them are equal and such that the sum of the four integers is \(12p+1\) is \(p(18p-7)\).