Standard +0.3 This is a standard dimensional analysis problem requiring students to equate dimensions of energy [ML²T⁻²] with those of m^a r^b ω^c, then solve three simultaneous equations for the exponents. While it requires careful bookkeeping and understanding of dimensions, it's a routine technique taught explicitly in Further Mechanics with no novel insight required.
3 A disc, of mass \(m\) and radius \(r\), rotates about an axis through its centre, perpendicular to the plane face of the disc.
The angular speed of the disc is \(\omega\).
A possible model for the kinetic energy \(E\) of the disc is
$$E = k m ^ { a } r ^ { b } \omega ^ { c }$$
where \(a , b\) and \(c\) are constants and \(k\) is a dimensionless constant.
Find the values of \(a , b\) and \(c\).
3 A disc, of mass $m$ and radius $r$, rotates about an axis through its centre, perpendicular to the plane face of the disc.
The angular speed of the disc is $\omega$.\\
A possible model for the kinetic energy $E$ of the disc is
$$E = k m ^ { a } r ^ { b } \omega ^ { c }$$
where $a , b$ and $c$ are constants and $k$ is a dimensionless constant.\\
Find the values of $a , b$ and $c$.\\
\hfill \mbox{\textit{AQA Further Paper 3 Mechanics 2019 Q3 [3]}}