Problems

A thin straight tube \(AB\) is rotated in a horizontal plane with uniform angular velocity \(\omega\) about an axis through \(A\) perpendicular to the axis of the tube. The position of the tube is defined by the angle \(\theta\) which its axis makes with some fixed axis in the plane of rotation. When \(\theta=0\), a smooth particle of mass \(m\) is started along the tube from \(A\) with a velocity \(u\). Show that the equation of the path followed by the particle is given by \[ r = \frac{u}{\omega} \sinh\theta, \] where \(r\) is its distance from \(A\). Show also that the torque required to maintain the motion of the tube is given by \[ mu^2 \sinh 2\theta. \]

Interpret the equation \[ S + \lambda t^2 = 0, \] where \(S=0\) and \(t=0\) are the equations of a conic and one of its tangents, \(\lambda\) being a parameter. Two chords \(AB\) and \(CD\) of a conic \(S\) meet in the point \(O\), and one of the tangents \(OP\) from \(O\) to \(S\) touches \(S\) at \(P\). Another conic \(S'\) is drawn through \(A, B, C, D\) to touch at \(P'\) the harmonic conjugate \(OP'\) of \(OP\) with respect to the line pair \(AB\) and \(CD\). Prove that there exists a conic \(S''\) having four point contact with \(S\) at \(P\) and four point contact with \(S'\) at \(P'\).

Draw a sketch of the curve \[ y^2 \frac{a^2-x^2}{c^2} = \frac{x^2}{b^2-x^2}, \] where \(a, b, c\) are positive and \(a < b\). Find the volumes of the solids obtained by the revolution of the loop about (i) the \(x\)-axis and (ii) the \(y\)-axis.

Shew that a system of coplanar forces can be uniquely reduced to three forces acting along the sides of an arbitrarily chosen triangle situated in the plane of the forces. Forces of magnitudes 1, 4, 2, 2, 6, 4 act along the sides \(AB, CB, CD, ED, FE, FA\) of a regular hexagon. Find their resultant and replace them by three forces acting along the sides of the triangle formed by \(AB, CD, EF\).

A long ladder of negligible weight rests with one end on a smooth horizontal plane and with the other projecting over the top of a smooth wall of height \(h\). A light inextensible cord is fastened at one end to the ladder and at the other to the foot of the wall, so that the ladder and cord make the same angle \(\theta (< \frac{1}{4}\pi)\) with the horizontal. The ladder and the cord are in a vertical plane at right angles to the wall. Shew that, if a man ascends the ladder, he will be able to reach a distance \[ h(\operatorname{cosec}\theta + \tfrac{1}{2}\tan 2\theta \sec\theta) \] along the ladder. If the plane were perfectly rough and the cord were removed, shew that he would only be able to ascend the smaller distance \[ h(\operatorname{cosec}\theta + \tan\theta\sec\theta). \]

Find the centre of gravity of a uniform solid hemisphere. A solid consists of a hemisphere of radius \(a\) from which a sphere of radius \(\frac{1}{2}a\) has been removed. It rests with its base on a rough plane inclined at an angle \(\alpha\) to the horizontal, and a gradually increasing force is applied at the pole of the hemisphere in a direction up the plane parallel to a line of greatest slope. Shew that the solid slips or tilts according as \(3\tan\lambda+2\tan\alpha\) is less than or greater than 3, where \(\lambda\) is the angle of friction. Find the value of the force required to destroy the equilibrium.

A framework consists of six equal rods freely jointed together to form a regular hexagon \(ABCDEF\), together with two struts \(FB, EC\). All the rods and struts have the same weight \(W\), and the framework is suspended from \(A\). Shew that the forces acting on the ends of the struts \(FB, EC\) are \(W\sqrt{37}\) and \(W\) respectively.

A gun is placed on a hillside which is in the form of a plane inclined at an angle \(\alpha\) to the horizontal. If the maximum muzzle velocity is \(V\), shew that the area within range is \[ \frac{\pi V^4}{g^2 \cos^2\alpha}. \]

Shew that Newton's experimental law connecting the relative velocity of two bodies before and after impact is equivalent to the statement that the ratio of the impulse of restitution to the impulse of compression is equal to the coefficient of restitution. Two equal smooth spheres are in contact on a smooth table. An impulse is applied to one sphere in a horizontal line through its centre. Shew that the maximum angle between the direction of motion of this sphere and the direction of the impulse is \[ \tan^{-1}\left(\frac{1+e}{2\sqrt{2}(1-e)}\right), \] where \(e\) is the coefficient of restitution. It is assumed that the initial impulse has ceased before the impulse of restitution comes into play.

A particle moves in a straight line, the relation between time and distance being \[ t = ax + bx^2, \] where \(a\) and \(b\) are constants. Determine the relations between (i) distance and acceleration, (ii) distance and velocity, (iii) time and velocity. If the particle travels 2000 feet in 1.9 seconds, and its velocity is then 1000 feet per second, shew that its initial velocity was about 1111 feet per second.