Standard +0.3 Part (a) is a standard dimensional analysis exercise requiring routine application of the method to find unknown exponents—typical M3 material but straightforward. Part (b) is a standard energy conservation problem with elastic potential energy. Both parts follow textbook patterns with no novel insight required, making this slightly easier than average.
Write down the dimensions of density, kinetic energy and power.
A sphere of radius \(r\) is moved at constant velocity \(v\) through a fluid.
In a viscous fluid, the power required is \(6 \pi \eta r v ^ { 2 }\), where \(\eta\) is the viscosity of the fluid.
Find the dimensions of viscosity.
In a non-viscous fluid, the power required is \(k \rho ^ { \alpha } r ^ { \beta } v ^ { \gamma }\), where \(\rho\) is the density of the fluid and \(k\) is a dimensionless constant.
Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
A rock of mass 5.5 kg is connected to a fixed point O by a light elastic rope with natural length 1.2 m . The rock is released from rest in a position 2 m vertically below O , and it next comes to instantaneous rest when it is 1.5 m vertically above O .
Find the stiffness of the rope.
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\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the dimensions of density, kinetic energy and power.
A sphere of radius $r$ is moved at constant velocity $v$ through a fluid.
\item In a viscous fluid, the power required is $6 \pi \eta r v ^ { 2 }$, where $\eta$ is the viscosity of the fluid.
Find the dimensions of viscosity.
\item In a non-viscous fluid, the power required is $k \rho ^ { \alpha } r ^ { \beta } v ^ { \gamma }$, where $\rho$ is the density of the fluid and $k$ is a dimensionless constant.
Use dimensional analysis to find $\alpha , \beta$ and $\gamma$.
\end{enumerate}\item A rock of mass 5.5 kg is connected to a fixed point O by a light elastic rope with natural length 1.2 m . The rock is released from rest in a position 2 m vertically below O , and it next comes to instantaneous rest when it is 1.5 m vertically above O .
Find the stiffness of the rope.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI M3 2010 Q1 [18]}}