Standard +0.3 This is a standard M2 elastic strings question requiring integration proof of EPE formula, equilibrium calculation, and energy conservation. While multi-part with several steps, each component follows routine procedures taught in M2 with no novel problem-solving required. The integration in part (a) is straightforward, and parts (b)(i-iii) apply standard formulas and energy methods that are well-practiced at this level.
An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\).
Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
A particle, of mass 5 kg , is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(O\).
The string has natural length 1.6 m and modulus of elasticity 392 N .
Find the extension of the string when the particle hangs in equilibrium.
The particle is pulled down to a point \(A\), which is 2.2 m below the point \(O\).
Calculate the elastic potential energy in the string.
The particle is released when it is at rest at the point \(A\).
Calculate the distance of the particle from the point \(A\) when its speed first reaches \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
8
\begin{enumerate}[label=(\alph*)]
\item An elastic string has natural length $l$ and modulus of elasticity $\lambda$. The string is stretched from length $l$ to length $l + e$.
Show, by integration, that the work done in stretching the string is $\frac { \lambda e ^ { 2 } } { 2 l }$.
\item A particle, of mass 5 kg , is attached to one end of a light elastic string. The other end of the string is attached to a fixed point $O$.
The string has natural length 1.6 m and modulus of elasticity 392 N .
\begin{enumerate}[label=(\roman*)]
\item Find the extension of the string when the particle hangs in equilibrium.
\item The particle is pulled down to a point $A$, which is 2.2 m below the point $O$.
Calculate the elastic potential energy in the string.
\item The particle is released when it is at rest at the point $A$.
Calculate the distance of the particle from the point $A$ when its speed first reaches $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA M2 2013 Q8 [13]}}