Problems

The two parabolas \[ y^2 = 4ax, \quad y^2=4bx, \] are drawn, where \(a\) and \(b\) are both positive; lines are then drawn through their foci perpendicular to their common axis. Find the area between the two parabolas which is contained between these lines; and shew that the volume generated by a revolution of this area round the axis is \(2\pi (a-b)^2(a+b)\).

A particle \(P\) is attracted towards each of four points \(A, B, C, D\) by forces equal to \(\mu_1 PA, \mu_2 PB, \mu_3 PC, \mu_4 PD\). Shew that it will rest in equilibrium only at the centre of gravity of four masses proportional to \(\mu_1, \mu_2, \mu_3, \mu_4\) placed at \(A, B, C, D\). Shew also that if a force \(Q\) be applied to the particle its position of equilibrium will be displaced from this position a distance \(Q/(\mu_1+\mu_2+\mu_3+\mu_4)\) in the direction of \(Q\).

A solid hemisphere rests with its base in an inclined position at an angle \(\theta\) to the horizontal, its curved surface resting on a horizontal plane (coefficient of friction \(\mu\)) and against a vertical plane (coefficient of friction \(\mu'\)). If the hemisphere is on the point of slipping shew that \[ \frac{c \sin\theta}{a} = \frac{\mu(1+\mu')}{1+\mu\mu'}, \] where \(a\) is the radius of the hemisphere and the centre of gravity is on the axis of symmetry at a distance \(c\) from the centre. If \(\mu=\mu'\) and \(c/a = 3/8\), shew that there is no position of limiting equilibrium for the hemisphere if \(5\mu > \sqrt{31}-4\).

\(OA\) is a slightly compressible vertical rod of height \(h\) and negligible mass (modulus of compressibility \(\mu\)) freely pivoted at its lowest point \(O\). \(AB\) is a slightly extensible cord of natural length \(l\) (modulus \(\lambda\)). \(B\) is a point in the horizontal plane through \(O\) distant \(a\) from \(O\) where \(a^2=l^2-h^2\). A horizontal force \(P\) is applied at \(A\) in the direction \(BO\). Shew that the horizontal and vertical components of the displacement of \(A\) are approximately (neglecting \(x^2\) and \(y^2\)) \[ x=P\left(\frac{h^3}{a^2\mu} + \frac{l^3}{a^2\lambda}\right), \quad y=\frac{Ph^2}{a\mu}. \]

A number of weights are to be hung on a light string so that the vertical lines drawn through them are at equal horizontal distances apart and so that the particles lie on a curve of the form \(a^2y=x^3\), where \(Ox\) is a horizontal axis and \(Oy\) an upward vertical axis (only positive values of \(x\) are to be considered). Shew that the weights of the particles must be in arithmetical progression.

Shew that the resultant of a number of parallel forces at a number of fixed points acts through a certain point which remains fixed if the direction of the parallel forces is changed, the magnitudes being unaltered. Explain the bearing of this theorem on the existence of a centre of gravity and prove in turn:

1. [(i)] that the centre of gravity of a uniform triangular plate is at the intersection of the medians,
2. [(ii)] that the centre of gravity of a uniform solid tetrahedron is at the point dividing the line joining a vertex to the centroid of the opposite face in the ratio 3:1,
3. [(iii)] that the same is true of a uniform solid cone with a plane base,
4. [(iv)] that the centre of gravity of a uniform hemispherical shell bisects the radius about which the shell is symmetrical,
5. [(v)] that the centre of gravity of a uniform solid hemisphere is at a distance from the centre equal to \(\frac{3}{8}\) of the radius.

Prove that the small oscillations of the bob of a simple pendulum are harmonic and that the time of swing is \(2\pi\sqrt{(l/g)}\). Assuming that the attraction of the earth varies inversely as the square of the distance from the centre of the earth at points outside and directly as the first power of the distance for points inside, and taking the radius of the earth as 4000 miles, shew that a pendulum clock would lose about 4.1 secs. per day at 1000 feet above the earth's surface and half that amount at 1000 feet below the surface.

Show that the least velocity (\(v\)) required to project a particle over a wall the top of which is at a height \(y\) above the point of projection and at a distance \(r\) from the point of projection is given by \[ v^2 = g(y+\sqrt{y^2+r^2}), \] and shew that \(u\), the horizontal component of the velocity, is given by \[ u^2 = \frac{1}{2}g(\sqrt{y^2+r^2}-y). \] Show that the particle must reach its maximum height before passing the top of the wall.

A projectile of mass \(m\) is fired horizontally with velocity \(u\) into a block of mass \(M\) which rests on a rough horizontal plane (coefficient of friction \(\mu\)). There is a horizontal force opposing penetration equal to \(R\). If \(l\) is the total distance which the projectile enters into the block and \(x\) is the distance travelled by the block before coming to rest, shew that \begin{align*} l &= \frac{ku^2}{2f'}, \\ x &= \frac{k^2u^2}{2}\left(\frac{1}{\mu g} - \frac{1}{f}\right), \end{align*} where \(k=m/(m+M)\) and \(R-\mu mg = Mf\) and provided that \(f\) is greater than \(\mu g\).

A point moves in a straight line with a retardation equal to \(kv^{n+1}\) where \(v\) is its velocity, and \(k, n\) are positive constants. If \(u\) is the velocity at time \(t=0\) shew that \[ v = u(1+nku^nt)^{-1/n}. \] Obtain the corresponding formula (i) when \(n=0\), (ii) when \(n\) lies between \(0\) and \(-1\), shewing that only in the last case will the point come to rest in a finite time.