Problems

State the laws of friction and explain what is meant by the angle of friction. A uniform rod rests in limiting equilibrium with its ends on a rough circular band whose plane is vertical. Prove that its inclination (\(\theta\)) to the vertical is given by the equation \[ \sin\lambda\sin(\theta+\lambda) = \cos\theta\cos^2\alpha, \] where \(\lambda\) is the angle of friction and \(2\alpha\) the angle subtended by the rod at the centre of the circle.

Define acceleration. The acceleration of a moving point decreases uniformly with the time; its value is \(f_1\) after a time \(t_1\) and \(f_2\) after a time \(t_2\) from the start. After what time will it have no acceleration? Shew also, that if its initial velocity is zero, the space described before its acceleration is zero is equal to \[ \frac{1}{3} \frac{(f_1t_2-f_2t_1)^3}{(t_2-t_1)(f_1-f_2)^2}. \]

A particle is projected under gravity with velocity \(u\) at an elevation \(\alpha\) to the horizon. Find the magnitude and direction of its velocity after a time \(t\). Shew that after a time \(\dfrac{u}{g}\csc\alpha\) its direction will be at right angles to its direction of projection, and that its distance from the point of projection will be equal to its depth below a horizontal line at a height \(\dfrac{u^2}{g}\) above the point of projection.

Explain what is meant by Simple Harmonic Motion and find the period. An elastic string hangs vertically from a fixed point. To the lower end is attached a heavy particle, which is then allowed to fall. When the particle reaches its lowest point half of it drops off. Shew that the other half will rise to a height \(2a\) above the starting point, where \(a\) is the extension of the string which the heavy particle would produce when hanging at rest.

Prove that, in a triangle \(ABC\), if \(x,y,z\) are the lengths of the perpendiculars from \(A,B,C\) on the opposite sides and \(R,r\) the radii of the circumscribed and inscribed circles,

1. [(i)] \(\dfrac{1}{R} = \dfrac{\cos A}{x} + \dfrac{\cos B}{y} + \dfrac{\cos C}{z}\),
2. [(ii)] \(\dfrac{1}{r} = \dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\),
3. [(iii)] \(4(R+r) = a^2\left(\dfrac{1}{y^2}+\dfrac{1}{z^2}-\dfrac{1}{x^2}\right)+b^2\left(\dfrac{1}{z^2}+\dfrac{1}{x^2}-\dfrac{1}{y^2}\right)+c^2\left(\dfrac{1}{x^2}+\dfrac{1}{y^2}-\dfrac{1}{z^2}\right)\).

Prove that \[ \cos \frac{A}{2} = \pm \frac{1}{2}(1+\sin A)^{\frac{1}{2}} \pm \frac{1}{2}(1-\sin A)^{\frac{1}{2}}, \] and determine the signs of the radicals when \(A\) is \(500^\circ\). If \(\sin 4\theta = m\), shew that the four values of \(\tan\theta\) are given by \[ \frac{1}{m}[\pm(1+m)^{\frac{1}{2}}-1][\pm(1-m)^{\frac{1}{2}}+1]. \]

Sum to \(n\) terms the series

1. [(i)] \(\cos\alpha\sin 2\alpha + \cos 2\alpha \sin 3\alpha + \cos 3\alpha \sin 4\alpha + \dots\),
2. [(ii)] \(\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{1}{7}+\tan^{-1}\frac{1}{13}+\tan^{-1}\frac{1}{21}+\dots\).

Express \(x^{2n}-2a^n x^n \cos n\theta + a^{2n}\) as the product of \(n\) real quadratic factors. A regular polygon of \(n+1\) sides is inscribed in a circle whose radius is \(a\). Prove that the product of all the lines joining one angular point of the polygon to the others is \((n+1)a^n\).

Shew that the circles, whose equations are of the form \[ x^2+y^2+a=\lambda x, \] where \(\lambda\) may have any value, have a common radical axis. Shew that the polars of a given point with respect to each of these circles pass through a fixed point.

Find the condition that the straight line \(y-k=m(x-h)\) shall be a tangent to the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1; \] hence find the locus of \((h,k)\) when the two tangents through \((h,k)\) are at right angles. If \(p_1\) and \(p_2\) are the lengths of the perpendiculars from the origin on parallel tangents to the two ellipses, \[ \frac{x^2}{a^2+\mu_1}+\frac{y^2}{b^2+\mu_1}=1 \quad \text{and} \quad \frac{x^2}{a^2+\mu_2}+\frac{y^2}{b^2+\mu_2}=1, \] prove that \(p_1^2 - p_2^2 = \mu_1 - \mu_2\).