A given set of coplanar forces reduces to a single resultant force, and is such that the total moment about a point \(O\) is \(Q\), while the sums of the components parallel to two perpendicular lines \(Ox, Oy\) are \(X\) and \(Y\) respectively. Find the equation of the line of action of the resultant. Forces equal to 1, 4, 2, and 6 lb. weight act along the sides \(OB, BC, CD, DO\) respectively of a square \(OBCD\) with side of length \(a\). Find the magnitude of their resultant and obtain the equation of the line of action referred to \(OB\) and \(OD\) as coordinate axes.
Obtain an expression for the ratio of the tensions at the two ends of a rope wound round a post of uniform coefficient of friction when the rope is in limiting equilibrium. A sailor is holding a ship by means of a horizontal rope wound round a post of a wharf. The coefficient of friction is 1/3. Find the maximum force that can be exerted by the ship if the sailor is to exert a pull of not more than 100 lb. and the rope is wrapped \(2\frac{1}{2}\) times round the post.
A uniform solid elliptic cylinder is in stable equilibrium resting on a perfectly rough horizontal table. Show that no amount of loading along its highest generator will render it unstable if the eccentricity of the orthogonal cross-section exceeds a certain value, to be found.
Three particles are simultaneously projected under gravity \(g\) in different directions from the same point. Show that after a time \(t\) they are at the vertices of a triangle of area proportional to \(t^2\). If the initial velocities of two of the particles are in the same vertical plane and are of magnitudes \(u, v\) and elevations \(\alpha, \beta\) respectively, show that the plane of this triangle will pass through the point of projection after a time $$\frac{2uv\sin(\beta - \alpha)}{g(u\cos\alpha - v\cos\beta)},$$ assuming this to be positive.
Solution: Suppose the positions of the particles are \(\mathbf{x}_1, \mathbf{x}_2,\mathbf{x}_3\), and \(\mathbf{g} = \begin{pmatrix} 0 \\ 0 \\ -g \end{pmatrix}\) relative to the initial point as the origin. \begin{align*} && \mathbf{x}_1 &= \mathbf{u}_1t - \frac12 \mathbf{g} t^2 \\ && \mathbf{x}_2 &= \mathbf{u}_2t - \frac12 \mathbf{g} t^2 \\ && \mathbf{x}_3 &= \mathbf{u}_3t - \frac12 \mathbf{g} t^2 \\ \end{align*} Then, \begin{align*} A &= |\frac12 (\mathbf{u}_2t - \frac12 \mathbf{g} t^2 - (\mathbf{u}_1t - \frac12 \mathbf{g} t^2)) \times (\mathbf{u}_3t - \frac12 \mathbf{g} t^2 - (\mathbf{u}_1t - \frac12 \mathbf{g} t^2))| \\ &= \frac12 |(\mathbf{u}_2t - \mathbf{u}_1t) \times (\mathbf{u}_3t -\mathbf{u}_1t )| \\ &= \frac{t^2}{2} |(\mathbf{u}_2 - \mathbf{u}_1) \times (\mathbf{u}_3-\mathbf{u}_1 )| \end{align*}
A smooth wire is in the form of one bay of a cycloid (with intrinsic equation \(s = 4a\sin\psi\)) vertical and concavity upwards. A bead of mass \(m\) slides down the wire from rest at the highest point under the action of gravity \(g\). Find the reaction of the wire when the particle reaches the point where the tangent is inclined at angle \(\psi\) to the horizontal. Show also that the magnitude of the resultant acceleration of the particle remains constant throughout the motion.
A uniform chain of total mass \(m\) and length \(l\) is released from rest when held vertically with its lower end just touching the bottom of the interior of a bucket of mass \(M\). When half of the chain has fallen into the bucket and lies coiled at the base, the bucket is the distance the bucket has fallen and \(y\) is the length of chain remaining above the base of the bucket. Show that the linear momentum of the system is given by \[ (M+m)\dot{x} - my\dot{y}/l \], and determine the velocity of the vertical part of the chain relative to the bucket at this stage in terms of \(y\).
(i) Obtain the expression $$\frac{1}{2}(m_1 + m_2)V^2 + \frac{1}{2}\frac{m_1m_2}{m_1 + m_2}v^2$$ for the kinetic energy of the translatory motion of two bodies of masses \(m_1\) and \(m_2\), where \(V\) is the velocity of the centre of mass and \(v\) the relative velocity of the centres of mass of the two bodies separately. (ii) A thin uniform tube of mass \(M\) rests on a smooth horizontal table, and within it is a smooth particle of mass \(m\) resting close to the middle point. If the tube is suddenly set rotating about its centre, which is initially at rest, show that when the particle leaves the tube the angular velocity will be reduced to \((M + m)/(M + 4m)\) of the initial value.
In the finite motion of a simple pendulum of length \(l\) under gravity \(g\), the inclination to the vertical oscillates between \(-\alpha\) and \(+\alpha\). Show that the total period of oscillation is given by $$4(l/g)^{1/2}\int_0^{\pi/2}(1 - \sin^2\frac{1}{2}\alpha\sin^2\psi)^{-1/2}d\psi.$$ If \(\alpha\) is sufficiently small for the integrand to be expanded in powers of \(\sin\frac{1}{2}\alpha\), show that to order \(\alpha^2\) the period is $$2\pi(l/g)^{1/2}(1 + \frac{1}{16}\alpha^2),$$ and find also the next term, in \(\alpha^4\).
A straight river of unit width is flowing with speed \(w\), and a swan starts and swims across, always endeavouring to reach the point \(O\) on the opposite bank directly opposite its starting point. The speed of the swan relative to the water is constant and equal to \(u\) (where \(u > w\)). If \(Or\) is taken along the river bank to correspond perpendicular to it across the river, show that the path of the swan has equation $$2x = y^{1-c} - y^{1+c},$$ where \(c = w/u\).
A groove of semicircular cross-section and radius \(b\) is cut round a right circular cylinder of radius \(a\) (where \(a > b\)). (The centres of the semicircular cross-sections lie in a circle perpendicular to the axis of the cylinder.) Show that the surface of the groove is \(2\pi^2ab - 4\pi b^2\). Find also the volume of material removed. Explain the relation between the two results.