OCR MEI M3 (Mechanics 3) 2008 January

1

1. 1. Write down the dimensions of force and the dimensions of density. When a wire, with natural length \(l _ { 0 }\) and cross-sectional area \(A\), is stretched to a length \(l\), the tension \(F\) in the wire is given by $$F = \frac { E A \left( l - l _ { 0 } \right) } { l _ { 0 } }$$ where \(E\) is Young's modulus for the material from which the wire is made.
   2. Find the dimensions of Young's modulus \(E\). A uniform sphere of radius \(r\) is made from material with density \(\rho\) and Young's modulus \(E\). When the sphere is struck, it vibrates with periodic time \(t\) given by $$t = k r ^ { \alpha } \rho ^ { \beta } E ^ { \gamma }$$ where \(k\) is a dimensionless constant.
   3. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. Fig. 1 shows a fixed point A that is 1.5 m vertically above a point B on a rough horizontal surface. A particle P of mass 5 kg is at rest on the surface at a distance 0.8 m from B , and is connected to A by a light elastic string with natural length 1.5 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-2_405_538_1338_845} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} The coefficient of friction between P and the surface is 0.4 , and P is on the point of sliding. Find the stiffness of the string.

2

1. A small ball of mass 0.01 kg is moving in a vertical circle of radius 0.55 m on the smooth inside surface of a fixed sphere also of radius 0.55 m . When the ball is at the highest point of the circle, the normal reaction between the surface and the ball is 0.1 N . Modelling the ball as a particle and neglecting air resistance, find
   1. the speed of the ball when it is at the highest point of the circle,
   2. the normal reaction between the surface and the ball when the vertical height of the ball above the lowest point of the circle is 0.15 m .
2. A small object Q of mass 0.8 kg moves in a circular path, with centre O and radius \(r\) metres, on a smooth horizontal surface. A light elastic string, with natural length 2 m and modulus of elasticity 160 N , has one end attached to Q and the other end attached to O . The object Q has a constant angular speed of \(\omega\) rad s \(^ { - 1 }\).
   1. Show that \(\omega ^ { 2 } = \frac { 100 ( r - 2 ) } { r }\) and deduce that \(\omega < 10\).
   2. Find expressions, in terms of \(r\) only, for the elastic energy stored in the string, and for the kinetic energy of Q . Show that the kinetic energy of Q is greater than the elastic energy stored in the string.
   3. Given that the angular speed of Q is \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tension in the string.

3 A particle is oscillating in a vertical line. At time \(t\) seconds, its displacement above the centre of the oscillations is \(x\) metres, where \(x = A \sin \omega t + B \cos \omega t\) (and \(A , B\) and \(\omega\) are constants).

1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \omega ^ { 2 } x\). When \(t = 0\), the particle is 2 m above the centre of the oscillations, the velocity is \(1.44 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, and the acceleration is \(0.18 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards.
2. Find \(A , B\) and \(\omega\).
3. Show that the period of oscillation is 20.9 s (correct to 3 significant figures), and find the amplitude.
4. Find the total distance travelled by the particle between \(t = 12\) and \(t = 24\).

4 Fig. 4.1 shows the region \(R\) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 8\). \begin{figure}[h] \end{figure}

1. Find the \(x\)-coordinate of the centre of mass of a uniform solid of revolution obtained by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.
2. Find the coordinates of the centre of mass of a uniform lamina in the shape of the region \(R\).
3. Using your answer to part (ii), or otherwise, find the coordinates of the centre of mass of a uniform lamina in the shape of the region (shown shaded in Fig. 4.2) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the line \(y = \frac { 1 } { 2 }\) and the line \(x = 1\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-4_595_1015_1610_607} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}