Question 1 - A-Level Maths

1. 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.
2. 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.
3. 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]
\includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-2_405_538_1338_845} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} 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.