| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle inside smooth hollow cylinder |
| Difficulty | Standard +0.8 This is a multi-part circular motion problem requiring force resolution in 3D (horizontal radial, vertical), understanding of normal reactions from two surfaces simultaneously, elastic string tension with Hooke's law, and finding a threshold condition. It goes beyond standard conical pendulum questions by involving contact forces from both base and curved surface, plus elastic string mechanics. The three-part structure with increasing complexity and the need to find a limiting condition elevates this above routine M2 questions. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.05b Circular motion: v=r*omega and a=v^2/r |
| Answer | Marks | Guidance |
|---|---|---|
| Vertical force \(= 10m\) | B1 | May be implied |
| \(10m = m v^2/0.4\) | M1 | Newton's Second Law radially |
| \(v = 2\) ms\(^{-1}\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(T = 13 \times (0.4 - 0.25)/0.25\) | B1 | \(T = 7.8\) N |
| \(m \times 8^2 \times 0.4 = 7.8 + 10m\) | M1 | Newton's Second Law radially, 2 horizontal forces |
| \(m = 0.5\) | A1 [3] | \(m(25.6 - 10) = 7.8\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(7.8 = m \times \omega^2 \times 0.4\) | M1 | Newton's Second Law radially, no horizontal reaction |
| \(\omega = 6.24\) | A1 [2] | \((\sqrt{39})\) |
**(i)**
| Vertical force $= 10m$ | B1 | May be implied |
| $10m = m v^2/0.4$ | M1 | Newton's Second Law radially |
| $v = 2$ ms$^{-1}$ | A1 [3] | |
**(ii)**
| $T = 13 \times (0.4 - 0.25)/0.25$ | B1 | $T = 7.8$ N |
| $m \times 8^2 \times 0.4 = 7.8 + 10m$ | M1 | Newton's Second Law radially, 2 horizontal forces |
| $m = 0.5$ | A1 [3] | $m(25.6 - 10) = 7.8$ |
**(iii)**
| $7.8 = m \times \omega^2 \times 0.4$ | M1 | Newton's Second Law radially, no horizontal reaction |
| $\omega = 6.24$ | A1 [2] | $(\sqrt{39})$ | [8] |4\\
\includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_170_616_1649_767}
A small sphere $S$ of mass $m \mathrm {~kg}$ is moving inside a fixed smooth hollow cylinder whose axis is vertical. $S$ moves with constant speed in a horizontal circle of radius 0.4 m and is in contact with both the plane base and the curved surface of the cylinder (see diagram).\\
(i) Given that the horizontal and vertical forces exerted on $S$ by the cylinder have equal magnitudes, calculate the speed of $S$.\\
$S$ is now attached to the centre of the base of the cylinder by a horizontal light elastic string of natural length 0.25 m and modulus of elasticity 13 N . The sphere $S$ is set in motion and moves in a horizontal circle with constant angular speed $\omega \mathrm { rads } ^ { - 1 }$ and is in contact with both the plane base and the curved surface of the cylinder.\\
(ii) It is given that the magnitudes of the horizontal and vertical forces exerted on $S$ by the cylinder are equal if $\omega = 8$. Calculate $m$.\\
(iii) For the value of $m$ found in part (ii), find the least possible value of $\omega$ for the motion.
\hfill \mbox{\textit{CAIE M2 2012 Q4 [8]}}