OCR MEI M3 2006 June — Question 1

Exam BoardOCR MEI
ModuleM3 (Mechanics 3)
Year2006
SessionJune
TopicDimensional Analysis

1
    1. Find the dimensions of power. In a particle accelerator operating at power \(P\), a charged sphere of radius \(r\) and density \(\rho\) has its speed increased from \(u\) to \(2 u\) over a distance \(x\). A student derives the formula $$x = \frac { 28 \pi r ^ { 3 } u ^ { 2 } \rho } { 9 P }$$
    2. Show that this formula is not dimensionally consistent.
    3. Given that there is only one error in this formula for \(x\), obtain the correct formula.
  1. A light elastic string, with natural length 1.6 m and stiffness \(150 \mathrm { Nm } ^ { - 1 }\), is stretched between fixed points A and B which are 2.4 m apart on a smooth horizontal surface.
    1. Find the energy stored in the string. A particle is attached to the mid-point of the string. The particle is given a horizontal velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to AB (see Fig. 1.1), and it comes instantaneously to rest after travelling a distance of 0.9 m (see Fig. 1.2). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-2_524_305_1274_639} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-2_524_305_1274_1128} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
    2. Find the mass of the particle.