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 \[ \sin^{-1}\left\{\frac{ac(c+2b)}{4b^2(c+b)}\right\}. \]
A bullet of mass 1 oz. is fired into a block of wood of mass 20 lb. which is suspended by a long string. The bullet becomes embedded in the block and the centre of mass of block and bullet rises to a height of 12 in. above its original position. What is the velocity of the bullet? \par If the resistance to the penetration of the bullet is \(675 v x^{1/2}\) lb. wt., where \(v\) is the velocity and \(x\) the penetration, shew that the final value of \(x\) is about 1 ft. The movement of the block during penetration may be neglected.
If a particle is moving in a curve, \(v\) being its velocity and \(\psi\) the angle between the direction of motion and a fixed direction, shew that the components of acceleration along the tangent and normal are \(\dfrac{dv}{dt}\) and \(v\dfrac{d\psi}{dt}\). \par If a particle moves on a rough inclined plane, the coefficient of friction being \(\mu\) and the inclination of the plane to the horizontal \(\alpha\), shew that the motion satisfies the equations \begin{align*} \frac{dv}{dt} &= -\mu g \cos\alpha + g \sin\alpha \cos\psi, \\ v\frac{d\psi}{dt} &= -g\sin\alpha \sin\psi, \end{align*} \(\psi\) being measured from the direction down a line of greatest slope. Hence shew that \[ v \sin^\mu\psi = C \left(\tan\frac{\psi}{2}\right)^{\cot\alpha}, \] where C is a constant.
ABCD is a rhombus of freely hinged light rods each of length \(l\). It is pivoted at A at a fixed point and C is connected to a point E vertically below A at a distance \(b\) from A by a vertical spring of natural length \(a\) and modulus of elasticity \(\lambda\); the lengths \(a, b, l\) are such that \(b>a+2l\). Two particles of mass \(m\) are attached at B and D and the whole rotates about AE with angular velocity \(\omega\). \par Shew that, if \(\omega^2 > \dfrac{g}{l} + \dfrac{\lambda(b-a-2l)}{mal}\), there is a position of relative equilibrium in which the rods make with the vertical an angle whose cosine is \[ \frac{mga+\lambda(b-a)}{l(ma\omega^2+2\lambda)}. \]
Prove that, if a number of forces act on a rigid body, the sum of them is equal to the mass multiplied by the acceleration of the centre of mass. \par Prove also that for a lamina rotating about a fixed axis perpendicular to it the sum of the moments of the forces about the axis is equal to the moment of inertia about the axis multiplied by the angular acceleration. \par A uniform circular disc of mass \(m\) and radius \(r\) is suspended with its plane vertical by pegs passing through two small holes drilled perpendicular to the plane of the disc, each distant \(a\) from the centre of the disc and distant \(d\) from each other at the same level above the centre of the disc. If one peg suddenly collapses, shew that the horizontal reaction at the other peg becomes immediately \[ \frac{mgd\sqrt{4a^2-d^2}}{4a^2+2r^2}. \]