| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Elastic string – horizontal circle on surface |
| Difficulty | Standard +0.3 This is a standard circular motion problem with elastic strings requiring application of Hooke's law and centripetal force equations. Part (i) involves straightforward substitution into F=mrω² and solving for extension. Part (ii) requires forming a general expression and considering the constraint that extension must be non-negative, which is routine for M2 level. The problem is slightly above average difficulty due to the two-part structure and the need to interpret physical constraints, but uses standard techniques without requiring novel insight. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.05b Circular motion: v=r*omega and a=v^2/r |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(T = 6e / 0.3\), \(0.2 \times 5^2(0.3 + e) = 6e / 0.3\), \(e = 0.1\) | B1, M1, A1, A1 | Newton's Second Law radially [4] |
| (ii) \(0.2\omega^2(0.3 + e) = 6e / 0.3\), \(e = 0.06\omega^2 / (20 - 0.2\omega^2)\), \(20 - 0.2\omega^2 > 0\), \((0 <) \omega < 10\) | M1, A1, M1, A1 | Newton's Second Law radially; Other forms acceptable; Uses denominator > 0; Disregard lower limit [4] |
**(i)** $T = 6e / 0.3$, $0.2 \times 5^2(0.3 + e) = 6e / 0.3$, $e = 0.1$ | B1, M1, A1, A1 | Newton's Second Law radially [4]
**(ii)** $0.2\omega^2(0.3 + e) = 6e / 0.3$, $e = 0.06\omega^2 / (20 - 0.2\omega^2)$, $20 - 0.2\omega^2 > 0$, $(0 <) \omega < 10$ | M1, A1, M1, A1 | Newton's Second Law radially; Other forms acceptable; Uses denominator > 0; Disregard lower limit [4]
---5 One end of a light elastic string of natural length 0.3 m and modulus of elasticity 6 N is attached to a fixed point $O$ on a smooth horizontal plane. The other end of the string is attached to a particle $P$ of mass 0.2 kg , which moves on the plane in a circular path with centre $O$. The angular speed of $P$ is $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$.\\
(i) For the case $\omega = 5$, calculate the extension of the string.\\
(ii) Express the extension of the string in terms of $\omega$, and hence find the set of possible value of $\omega$.
\hfill \mbox{\textit{CAIE M2 2011 Q5 [8]}}