Problems

State the laws of statical friction and find the least force that will support a heavy particle in equilibrium when placed on an inclined plane whose inclination to the horizontal is greater than the angle of friction. A rectangular block on a square base of side \(a\) is of height \(b\) and rests on a rough inclined plane of inclination \(\alpha\) so that two faces are vertical. A string is attached to the middle point of the highest edge and is pulled up the plane in a direction parallel to a line of greatest slope. Prove that, as the tension of the string is gradually increased, the equilibrium of the block will be broken by sliding if \(\mu < (a-b\tan\alpha)/2b\), where \(\mu\) is the coefficient of friction between the block and the face of the plane.

Prove that, (i) the centre of inertia of a uniform triangular lamina is the same as that of three equal particles placed at the angular points, (ii) the centre of inertia of a uniform quadrilateral lamina is the same as that of four equal particles placed at the angular points and of an equal negative particle placed at the intersection of the diagonals. Find the ratio of the masses of three particles placed at the angular points of a triangle such that their centre of inertia is at the orthocentre of the triangle.

State the principle of conservation of linear momentum. A wedge of mass \(M\) whose faces are each inclined at an angle of 45° to the horizontal rests with its base on a smooth horizontal plane and is free to move in a direction perpendicular to its edge. Particles of masses \(m, m'\) connected by a light string passing over the edge are placed one on each face of the edge with the string taut. Prove that when the system is released from rest the acceleration of the wedge is \[ (m \sim m')g/(2M+m+m'). \]

Prove that the acceleration towards the centre of a particle moving in a circle is \(v^2/r\). Two particles describe two circles in a plane uniformly in the same time. Prove that the acceleration of one relative to the other is constant in magnitude and changes its direction uniformly.

A particle is moving in a straight line under a force to a fixed point in the line proportional to the distance from the point. Prove that the motion is simple harmonic and find the period. Two light elastic strings of natural lengths \(l, l'\) and moduli \(E, E'\) respectively are knotted together to form one string, one end of which is fixed while the other is attached to a particle of mass \(m\) which oscillates freely in a vertical line under the action of gravity and the tension of the string. Prove that the period of an oscillation is the same as that of a simple pendulum of length \(mg(l/E+l'/E')\).

Solve the equations

1. [(i)] \(x(y+z) = y(z+x) = z(x+y) = a^2\),
2. [(ii)] \(x+y+z=2, \quad x^2+y^2+z^2=26, \quad x^3+y^3+z^3=38\).

Find the conditions that

1. [(i)] \(ax^2+2bxy+cy^2\).
2. [(ii)] \(ax^2+2hxy+by^2+2gx+2fy+c\)

should be positive for all real values of \(x\) and \(y\). Find the condition that \((x-a)/(x-b)(x-c)\) may be capable of taking all real values.

Prove that the arithmetic mean of \(n\) positive quantities is not less than their geometric mean. Show that the sum of a series of \(n\) positive terms forming a harmonical progression is not less than \(n\) times the harmonic mean between the first and last terms.

Obtain the expansion of \(\log_e(1+x)\) from the exponential theorem. Prove that the sum to infinity of the series \[ \frac{1}{1(p+1)} + \frac{1}{2(p+2)} + \frac{1}{3(p+3)} + \dots \] is \[ \frac{1}{p}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{p}\right). \]

Prove the law of formation of the successive convergents of the continued fraction \[ \frac{1}{a_1+} \frac{1}{a_2+} \frac{1}{a_3+} \dots. \] Prove that \[ \frac{1}{2+} \frac{2}{3+} \frac{3}{4+} \dots \text{ to infinity} \] is equal to \(\frac{3-e}{e-2}\).