| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Horizontal elastic string on rough surface |
| Difficulty | Standard +0.8 This M3 question requires energy methods with elastic strings including friction work, finding modulus of elasticity from given tension, and solving a two-part motion problem where the particle passes through natural length then comes to rest. It demands careful bookkeeping of multiple energy terms and understanding of when friction acts, making it moderately challenging but within standard M3 scope. |
| Spec | 3.03r Friction: concept and vector form3.03v Motion on rough surface: including inclined planes6.02g Hooke's law: T = k*x or T = lambda*x/l6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings |
(a)
M1 for attempting Hooke's Law. Formula must be correct, either explicitly or by correct substitution.
A1 for $20 = \frac{\lambda \times 0.3}{1.2}$
A1 for obtaining $\lambda = 80$
B1 for the initial EPE $= 3$ J (their value for $\lambda$ allowed). May only be seen in the equation.
M1 for a work-energy equation with one EPE term, one KE term and work done against friction. Award if second EPE/KE terms included provided these become 0. The EPE must be dimensionally correct, but need not be fully correct (e.g., denominator 1.2 instead of 2.4).
A1ft for a completely correct equation, follow through their EPE.
A1 cao for $v = 0.80$ or $0.805$ m s$^{-1}$. Must be 2 or 3 sf.
Note: This is damped harmonic motion (due to friction) so all SHM attempts lose the last 4 marks.
(b)
M1 for any complete method leading to a value for either BC. If the distance travelled after the string becomes slack is found, the work must be completed by adding 0.3. Their EPE found in (a) used in energy methods.
MS method is energy from B to C, i.e., work done against friction = loss of EPE.
OR Energy from point where the string becomes slack to C, i.e., work done against friction = loss of KE and completed for the required distance.
OR Newton's second law to obtain the acceleration $-\frac{2g}{5}$ while the string is slack and $v^2 = u^2 + 2as$ to find the distance and completed for the required distance.
A1cso for $BC = 0.38$ or $0.383$ m. Must be 2 or 3 sf.
---\begin{enumerate}
\item A particle $P$ of mass 2 kg is attached to one end of a light elastic string of natural length 1.2 m . The other end of the string is attached to a fixed point $O$ on a rough horizontal plane. The coefficient of friction between $P$ and the plane is $\frac { 2 } { 5 }$. The particle is held at rest at a point $B$ on the plane, where $O B = 1.5 \mathrm {~m}$. When $P$ is at $B$, the tension in the string is 20 N . The particle is released from rest.\\
(a) Find the speed of $P$ when $O P = 1.2 \mathrm {~m}$.
\end{enumerate}
The particle comes to rest at the point $C$.\\
(b) Find the distance $B C$.\\
\hfill \mbox{\textit{Edexcel M3 2013 Q4 [9]}}