Standard +0.3 This is a standard dimensional analysis question with routine follow-up parts. Part (a)(ii) requires setting up and solving three simultaneous equations from dimensional homogeneity, which is a well-practiced technique. Parts (a)(iii) and (b) involve straightforward applications of the derived formula and standard SHM formulas (v_max = ωA, v² = ω²(A² - x²)). While multi-part and requiring several techniques, all steps are textbook-standard with no novel insight required, making it slightly easier than average.
Write down the dimensions of force.
The period, \(t\), of a vibrating wire depends on its tension, \(F\), its length, \(l\), and its mass per unit length, \(\sigma\).
Assuming that the relationship is of the form \(t = k F ^ { \alpha } l ^ { \beta } \sigma ^ { \gamma }\), where \(k\) is a dimensionless constant, use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\).
Two lengths are cut from a reel of uniform wire. The first has length 1.2 m , and it vibrates under a tension of 90 N . The second has length 2.0 m , and it vibrates with the same period as the first wire.
Find the tension in the second wire. (You may assume that changing the tension does not significantly change the mass per unit length.)
The midpoint M of a vibrating wire is moving in simple harmonic motion in a straight line, with amplitude 0.018 m and period 0.01 s .
Find the maximum speed of M .
Find the distance of M from the centre of the motion when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
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\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the dimensions of force.
The period, $t$, of a vibrating wire depends on its tension, $F$, its length, $l$, and its mass per unit length, $\sigma$.
\item Assuming that the relationship is of the form $t = k F ^ { \alpha } l ^ { \beta } \sigma ^ { \gamma }$, where $k$ is a dimensionless constant, use dimensional analysis to determine the values of $\alpha , \beta$ and $\gamma$.
Two lengths are cut from a reel of uniform wire. The first has length 1.2 m , and it vibrates under a tension of 90 N . The second has length 2.0 m , and it vibrates with the same period as the first wire.
\item Find the tension in the second wire. (You may assume that changing the tension does not significantly change the mass per unit length.)
\end{enumerate}\item The midpoint M of a vibrating wire is moving in simple harmonic motion in a straight line, with amplitude 0.018 m and period 0.01 s .
\begin{enumerate}[label=(\roman*)]
\item Find the maximum speed of M .
\item Find the distance of M from the centre of the motion when its speed is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI M3 2006 Q1 [18]}}