Standard +0.3 This is a standard dimensional analysis problem requiring students to equate dimensions of time with combinations of length, force, and mass per unit length. While it involves setting up and solving simultaneous equations from dimensional considerations, it follows a well-established algorithmic procedure taught explicitly in M3 with no novel insight required. Slightly above average difficulty due to the algebraic manipulation needed.
2 The time, \(t\), for a single vibration of a piece of taut string is believed to depend on
the length of the taut string, \(l\),
the tension in the string, \(F\),
the mass per unit length of the string, \(q\), and
a dimensionless constant, \(k\),
such that
$$t = k l ^ { \alpha } F ^ { \beta } q ^ { \gamma }$$
where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, find the values of \(\alpha , \beta\) and \(\gamma\).
2 The time, $t$, for a single vibration of a piece of taut string is believed to depend on\\
the length of the taut string, $l$,\\
the tension in the string, $F$,\\
the mass per unit length of the string, $q$, and\\
a dimensionless constant, $k$,\\
such that
$$t = k l ^ { \alpha } F ^ { \beta } q ^ { \gamma }$$
where $\alpha , \beta$ and $\gamma$ are constants.\\
By using dimensional analysis, find the values of $\alpha , \beta$ and $\gamma$.
\hfill \mbox{\textit{AQA M3 2011 Q2 [5]}}