A peg is fixed in a horizontal table and a lamina with a straight slot cut in it is placed on the table with the peg through the slot; if a given point of the slot is made to describe a straight line along the table, find the locus of the centre of instantaneous rotation (i) on the table, (ii) on the lamina.
Two vertical posts of heights \(a,b\) stand on level ground at a distance \(c\) apart; a stone is projected from the ground level with the least possible velocity consistent with its just clearing the two posts. Prove that the latus rectum of this parabolic trajectory is \(c^2/d\), where \(d^2=(a-b)^2+c^2\), and that the range on the ground level is \[ c\{d^2+2(a+b)d+(a-b)^2\}^{\frac{1}{2}}/2d. \]
A jointed framework \(ABCD\) consisting of four equal uniform rods is caused to rotate in a horizontal plane about the corner \(A\) with a given angular velocity. It is held rigid either (a) by a tie between \(C\) and \(A\) in which the tension is \(T\), or (b) by a light strut between \(B\) and \(D\) in which the thrust is \(P\). Show that \(\frac{T}{AC}\) in case (a) is equal to \(\frac{P}{BD}\) in case (b).
Find the cartesian equations for the smooth cycloid on which a particle will describe simple harmonic oscillations of period \(T\) under the action of gravity. Show that if the particle is projected horizontally at the lowest point with the velocity acquired in falling through a height equal to half the length of the cycloid from cusp to cusp it will take a time \(\frac{1}{4}T\) to reach the cusp.
Find expressions for the components of acceleration along and perpendicular to the radius vector of a point whose polar coordinates are known as functions of time. Find the law of force under which a particle can describe the spiral \(r=a\theta+b\), and find the velocity at any point of the path.
A uniform circular hoop of radius \(r\) rolls steadily on a horizontal plane so that its centre describes with velocity \(V\) a horizontal circle of radius \(R\). Its plane makes a constant angle \(\alpha\) with the horizontal. Prove that \[ V^2 = \frac{2gR^2\cot\alpha}{4R+r\cos\alpha}. \]
Prove that the small oscillations of a dynamical system about a position of equilibrium are compounded of a number of simple modes for each of which all coordinates of the system execute simple harmonic oscillations. Find the periods of oscillation in a vertical plane of a system consisting of two equal uniform rods \(AB, BC\) jointed together at \(B\) and hung from a joint at \(A\) so that they are vertical in their equilibrium position.
Explain the method of images in electrostatics. Two dielectrics of specific inductive capacity \(K_1\) and \(K_2\) are separated by an infinite plane face. Two charges \(e_1\) and \(e_2\) are placed at any points in the two media. Find the direction and magnitude of the force which acts on the charge \(e_1\). (The line joining \(e_1\) and \(e_2\) is not necessarily perpendicular to the interface.)
Define the coefficients of capacity \(q_{rs}\) for a system of conductors and show that \(q_{rs}=q_{sr}\). Find the coefficients of capacity for a system consisting of three concentric spheres of radii \(a,b,c\).
Prove that a solid gravitating sphere attracts external bodies as though its whole mass were concentrated at the centre. Gravitating matter is uniformly distributed between two concentric spherical surfaces of radii \(a\) and \(b\), (\(a>b\)). A particle starting from rest at the outer surface falls through a smooth hole into the hollow core, acquiring a velocity \(v\) in doing so. If \(c\) is the velocity acquired by a particle in falling from infinity to the outer surface show that \[ \left(\frac{v}{c}\right)^2 = \frac{a^2+2b^3-3ab^2}{a^3}. \]