| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Centre of mass with variable parameter |
| Difficulty | Standard +0.8 This is a multi-step centre of mass problem requiring calculation of composite shapes (cone + hollow cylinder), use of standard formulae for centres of mass, and application of the toppling condition. While methodical, it requires careful bookkeeping of multiple components and the toppling equilibrium condition adds problem-solving beyond routine exercises. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass |
| Answer | Marks | Guidance |
|---|---|---|
| 5(i) | tanθ = (0.6 – 0.5)/0.4 (= 1/4) | B1 |
| tanθ = x/0.6 | M1 | |
| x = 0.15 m AG | A1 | |
| Total: | 3 |
| Answer | Marks |
|---|---|
| 5(ii) | (π0.62 × 0.8/3) × (0.8/4) – [π(0.52 – x2) × 0.4] × (0.4/2) |
| = [π0.62 × 0.8/3 + π(0.52 – x2) × 0.4]x | M1 |
| A1 | Attempts to take moments about the base of the cone using their |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | Attempts to solve the equation | |
| x = 0.464 | A1 | Note x2 = 0.216 |
| Total: | 4 |
Question 5:
--- 5(i) ---
5(i) | tanθ = (0.6 – 0.5)/0.4 (= 1/4) | B1 | θ is the angle made by the base and the vertical
tanθ = x/0.6 | M1
x = 0.15 m AG | A1
Total: | 3
--- 5(ii) ---
5(ii) | (π0.62 × 0.8/3) × (0.8/4) – [π(0.52 – x2) × 0.4] × (0.4/2)
= [π0.62 × 0.8/3 + π(0.52 – x2) × 0.4]x | M1
A1 | Attempts to take moments about the base of the cone using their
x
Note x=0.15 Correct equation for the A mark.
M1 | Attempts to solve the equation
x = 0.464 | A1 | Note x2 = 0.216
Total: | 4\includegraphics{figure_5}
A uniform object is made by joining a solid cone of height 0.8 m and base radius 0.6 m and a cylinder. The cylinder has length 0.4 m and radius 0.5 m. The cylinder has a cylindrical hole of length 0.4 m and radius $x$ m drilled through it along the axis of symmetry. A plane face of the cylinder is attached to the base of the cone so that the object has an axis of symmetry perpendicular to its base and passing through the vertex of the cone. The object is placed with points on the base of the cone and the base of the cylinder in contact with a horizontal surface (see diagram). The object is on the point of toppling.
\begin{enumerate}[label=(\roman*)]
\item Show that the centre of mass of the object is 0.15 m from the base of the cone. [3]
\item Find $x$. [4]
\end{enumerate}
[The volume of a cone is $\frac{1}{3}\pi r^2 h$.]
\hfill \mbox{\textit{CAIE M2 2018 Q5 [7]}}