Standard +0.3 Part (a) is a standard dimensional analysis exercise requiring recall of basic dimensions and solving simultaneous equations for exponents—routine for M3 students. Part (b) is a straightforward statics problem with friction and elastic strings, requiring standard force resolution. Both parts follow textbook patterns with no novel insight required.
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.
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.
Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
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]
1
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\item 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.
\item 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.
\item Use dimensional analysis to find $\alpha , \beta$ and $\gamma$.
\end{enumerate}\item 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]
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\includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-2_405_538_1338_845}
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\caption{Fig. 1}
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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.
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\hfill \mbox{\textit{OCR MEI M3 2008 Q1 [18]}}