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:
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.
\(OA, AB\) are two inextensible strings each of length 5 ft. \(O\) is attached to a fixed point and masses \(m_1, m_2\) are attached at \(A\) and \(B\). The whole swings round so that both strings lie in a vertical plane which rotates about the vertical through \(O\) with uniform angular velocity \(\omega\). Shew that if \(A\) and \(B\) remain at distances 3 ft. and 7 ft. respectively from the axis of rotation the ratio \(m_1/m_2\) is equal to 49/15, and find the necessary value of \(\omega\).