Define a simple harmonic motion. Find the period of such a motion and shew that it is independent of the amplitude. A particle of weight \(w\) is attached to one end of a light elastic string of modulus \(\lambda\) and natural length \(l\), the other end of the string being attached to a fixed point. The weight is let go from rest from the fixed point. Describe the subsequent motion and find the time until the particle first returns to the fixed point.
Differentiate \(\sin^{-1}(\log\tan x)\). Find the \(n\)th differential coefficients of \[ \text{(1) } x^2e^{ax}, \quad \text{(ii) } \frac{1}{1+x^3}. \]
Explain how to find the maxima and minima values of a function of \(x\).
Find the values of \(x\) that give maximum or minimum values to \((x-a)^2(x-b)^3\), distinguishing the cases according as \(a>=
Prove that for a plane curve the radius of curvature \(\rho=r\frac{dr}{dp}\). Shew that the radius of curvature at a point of the curve \(r^n=a^n\cos n\theta\) is \(a^n/\{(n+1)r^{n-1}\}\).
Two equal and similar homogeneous cubical blocks each of weight \(W\) are smoothly hinged together along a common edge and rest symmetrically on a rough horizontal plane with the line of hinges horizontal and two edges in contact with the plane. The faces through these edges and the line of hinges each make an acute angle \(\tan^{-1} t\) with the vertical. Prove that, if \(\mu\), the coefficient of friction, is less than \(\frac{1}{2}\), \(t\) must lie between \(1-2\mu\) and \(1+2\mu\); but if \(\mu\) is greater than \(\frac{1}{2}\), \(t\) must lie between 0 and \(1+2\mu\). Shew that the least vertical force applied at the middle point of the common edge which will disturb equilibrium is \(W(2\mu+t-1)/(\mu+t)\) when the force is upwards and is \(W(2\mu-t+1)/(t-\mu)\) when the force is downwards and \(t>\mu\). What is the result in the latter case when \(t<\mu\)?
A particle is attached to a fixed point in a rough horizontal plane by means of an elastic string; the string is drawn out horizontally to its natural length and the particle is projected along the plane away from the fixed point with velocity \(u\). Find where the particle first comes to rest and shew that it will remain permanently at rest or return towards the fixed point according as \(u\) is less or greater than \(\mu(3gb)^{\frac{1}{2}}\), where \(\mu\) is the coefficient of friction, and \(b\) is the extension which would be produced by applying a statical force equal to the weight of the particle at the end of the elastic string.
Shew how to find by a graphical method the resultant of any number of coplanar forces. Forces of 2, 3 and 4 units act along the sides, taken in order, of an equilateral triangle, the length of a side being 3". Find graphically the magnitude and line of action of their resultant.
A uniform triangular lamina \(ABC\), right angled at \(A\) rests in a vertical plane with the sides \(AB, AC\) supported on two smooth pegs at \(D, E\) in a horizontal line at a distance \(\frac{1}{n} BC\) apart. Prove that the inclination \(\theta\) of \(BC\) to the horizontal is given by the equation \[ n\cos(\theta+2C) = 3\cos 2(\theta+C). \]
A circular disc of weight \(W\) and radius \(a\) is suspended horizontally by a number of vertical strings, each of length \(2a\), attached symmetrically to points on its circumference. Shew that the horizontal couple required to keep it raised through a distance \(h\) is \[ \frac{1}{2}W\sqrt{\{h(4a-h)\}}. \]
A uniform solid hemisphere of weight \(W\) and radius \(a\) rests with vertex downwards on a horizontal plane, and a particle of weight \(w\) is placed upon it at the centre of the sphere. A gradually increasing horizontal force is applied to a point of the rim until the hemisphere just slips on the plane. Prove that the particle will not slide on the upper face, provided that the coefficient of friction is \(> \frac{(W+w)a}{2Wh} - \frac{Wh}{2(W+w)a}\), where \(h\) is the distance of the centroid of the hemisphere from the centre.