Problems

Prove the formulae \(\ddot{r}-r\dot{\theta}^2\), \(r\ddot{\theta}+2\dot{r}\dot{\theta}\) for the radial and transverse components of acceleration in polar coordinates. \par A smooth small ring P is fixed to one edge of a smooth horizontal table. A light thread AB passes through P; to one end A, which is on the table, is fastened a mass \(m_1\), and to the other end B, which hangs below P, is attached a mass \(m_2\). If during the motion of the system \(m_1\) moves directly towards P and \(m_2\) moves in a vertical plane, write down the equations of motion of \(m_1\) and \(m_2\) in terms of \(r\), the distance of \(m_2\) from P, and \(\theta\), the angle that PB makes with the vertical. \par If at \(t=0\), when \(m_1\) is at rest and PB is vertical and of length \(l\), \(m_2\) is projected horizontally with velocity \(V\), find the initial values of \(\dot{r}\) and \(\dot{\theta}\) and the initial tension in the thread.

If a particle is describing a circle of radius \(a\) with constant speed \(v\), show that the acceleration is along the radius, and that its magnitude is \(v^2/a\). \par A cylindrical shaft is rotating about its axis, which is vertical, with constant angular velocity \(\omega\), and AOA' is a diameter of the shaft. Light rods AB, A'B' are freely pivoted to the shaft at A, A', and carry at their ends blocks B, B', each of mass \(m\), which slide against the rough inside surface of a fixed cylindrical drum. The drum surrounds the shaft and is co-axial with it; the plane containing AB, A'B' and O is perpendicular to the axis of the shaft, and the angles ABO, A'B'O are equal to \(\alpha\). Show that the couple exerted on the drum is \[ 2\mu m b^2 \omega^2 \sin\alpha / (\sin\alpha + \mu\cos\alpha), \] where \(b = OB\) and \(\mu\) is the coefficient of friction between the blocks and the drum. \par [Diagram of a rotating shaft with pivoted rods and blocks inside a drum is shown]. \par Investigate whether this result still holds when the shaft is rotating in the opposite direction to that shown in the diagram.

Spheres are described to touch two fixed planes and to pass through a fixed point. Prove that they all pass through a second fixed point and that the locus of their points of contact with either plane is a circle.

Evaluate \[ \int_0^1 \sqrt{\frac{1-x}{1+x}} dx, \quad \int_0^{\pi/4} \frac{x}{\cos^4 x} dx, \quad \int_0^\infty \frac{x\,dx}{x^3+1}. \]

State the laws of friction. \par Two particles of mass \(m\) lying on a rough horizontal table are connected by an elastic string which is stretched to such a length that the tension is \(kmg\) where \(k < \mu\) (the coefficient of friction). The table is then slowly tilted about a horizontal line parallel to the direction of the string. Shew that the particles will begin to slip when the inclination \(\alpha\) of the table is given by \(\tan^2 \alpha = (\mu^2-k^2)/(k^2+1)\), and that the directions in which they begin to slip will make angles \(\cot^{-1}(\sin\alpha/k)\) with the lines of greatest slope.

Find the position of the centre of mass of a uniform solid bounded by a parabolic cylinder of latus rectum \(4a\), by two planes perpendicular to the generators, and a plane perpendicular to the axis of symmetry at a distance \(h\) from the vertex. \par Shew that the solid will rest in equilibrium on a horizontal plane with its plane of symmetry inclined at an angle \(\tan^{-1}\sqrt{\frac{5a}{3h-10a}}\) to the horizontal provided that \(3h > 10a\).

ABCD, A'BC'D' are crossed light rods pivoted at B; \[ AB = A'B = 1\frac{1}{2}\text{ ft.},\quad BC=BC'=2\text{ ft.},\quad CD=C'D'=1\text{ ft.}; \] CC' are connected by a light rod of length 2 ft. pivoted at C and C'. \par The framework stands in a vertical plane on smooth ground at A' and D and a weight W is suspended from A and D' by strings of length 2 ft. and 3 ft. respectively. \par By graphical constructions find the pressures on the ground at A' and D and the reactions at the pivots B, C, C'.

A picture frame has eyelets in the back each at a distance 30 in. from the bottom of the frame and such that the points of attachment of the picture cords are 1 in. from the back of the frame. The centre of mass of the frame is 20 in. above the bottom and 1 in. in front of the back of the frame. The frame is hung against a smooth vertical wall by two equal parallel cords, each of length 40 in., attached to two hooks, the points of suspension being 1 in. in front of the wall. Write down equations for the small inclinations \(\theta, \phi\) of the cords and the back of the frame to the vertical. Shew that approximately \(\theta = 2/65\) and \(\phi=7/65\) in circular measure.

A light rigid platform AB rests horizontally in equilibrium on and is attached to a number of vertical slightly compressible springs, the extreme springs being at the ends A, B; a weight W is placed at a point P. Shew that whatever be the compressibilities of the various springs (not all equal to one another) there is a point X of the platform between A and B such that, wherever P may be, the tilt of the platform is proportional to PX.

The tractive force per unit weight of an electric train is given at velocity \(u\) by \[ \frac{a(c-u)}{(b+u)^2}, \] where \(a, b, c\) are constants. Find the speed at which the power exerted is a maximum; and shew that, neglecting road and air resistance, this speed will be the maximum obtainable up a gradient \[ \sin^{-1}\left\{\frac{ac(c+2b)}{4b^2(c+b)}\right\}. \]