| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Friction |
| Type | Elastic string with friction |
| Difficulty | Challenging +1.2 This is a multi-stage mechanics problem combining elastic strings, energy conservation, and friction. It requires setting up energy equations (considering elastic PE, gravitational PE, and KE), applying limiting equilibrium conditions, and resolving forces. While it involves several concepts and careful bookkeeping of the string extension at different positions, the individual techniques are standard M2 material with no novel insights required. The multi-part nature and integration of topics makes it moderately above average difficulty. |
| Spec | 3.03u Static equilibrium: on rough surfaces6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Ext} = 0.8 + 0.9 - 1.4\ (= 0.3\) m\()\) | B1 | Ext when in limiting equilibrium |
| \(EE = 70\times0.30^2/(2\times1.4)\ (= 2.25\) J\()\) | B1 | EE in limiting equilibrium |
| M1 | EE/PE/KE balance | |
| \(0.3v^2/2 = 0.3g\times0.8 - 2.25\) | A1 | |
| \(v = 1\) ms\(^{-1}\) | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(T = 70\times0.3/1.4\ (= 15\) N\()\) | B1 | Uses ext from part (i) |
| \(15 = \mu(3g)\) | M1 | \(F = \mu R\), using mass of B |
| \(\mu = 0.5\) | A1 [3] [8] |
## Question 5:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Ext} = 0.8 + 0.9 - 1.4\ (= 0.3$ m$)$ | B1 | Ext when in limiting equilibrium |
| $EE = 70\times0.30^2/(2\times1.4)\ (= 2.25$ J$)$ | B1 | EE in limiting equilibrium |
| | M1 | EE/PE/KE balance |
| $0.3v^2/2 = 0.3g\times0.8 - 2.25$ | A1 | |
| $v = 1$ ms$^{-1}$ | A1 [5] | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $T = 70\times0.3/1.4\ (= 15$ N$)$ | B1 | Uses ext from part (i) |
| $15 = \mu(3g)$ | M1 | $F = \mu R$, using mass of B |
| $\mu = 0.5$ | A1 [3] [8] | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_341_529_260_808}
A block $B$ of mass 3 kg is attached to one end of a light elastic string of modulus of elasticity 70 N and natural length 1.4 m . The other end of the string is attached to a particle $P$ of mass 0.3 kg . $B$ is at rest 0.9 m from the edge of a horizontal table and the string passes over a small smooth pulley at the edge of the table. $P$ is released from rest at a point next to the pulley and falls vertically. At the first instant when $P$ is 0.8 m below the pulley and descending, $B$ is in limiting equilibrium with the part of the string attached to $B$ horizontal (see diagram).\\
(i) Calculate the speed of $P$ when $B$ is first in limiting equilibrium.\\
(ii) Find the coefficient of friction between $B$ and the table.
\hfill \mbox{\textit{CAIE M2 2013 Q5 [8]}}