| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2012 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Work done and energy |
| Type | Particle on smooth curved surface |
| Difficulty | Standard +0.3 This is a straightforward energy conservation problem requiring students to apply work-energy principles in two parts. Part (i) involves a direct calculation using gravitational PE change and the work-energy theorem. Part (ii) requires setting up an equation with the given ratio of work done, but the method is standard. The geometry is given explicitly (no need to derive heights), and the problem follows a familiar textbook pattern for M1 level. |
| Spec | 6.02d Mechanical energy: KE and PE concepts |
| Answer | Marks | Guidance |
|---|---|---|
| (i) PE loss = \(0.8g \times (2.5 - 1.8)\) (= 5.6J) | B1 | |
| Work done is 5.6 J | B1 | 2 marks total |
| (ii) \(\frac{1}{2} \times 0.8v^2 = 0.8g \times 2.5 - 0.6 \times 5.6\) | A1ft | |
| Speed at \(B\) is 6.45 m s\(^{-1}\) | A1 | 3 marks total |
**(i)** PE loss = $0.8g \times (2.5 - 1.8)$ (= 5.6J) | B1 |
Work done is 5.6 J | B1 | 2 marks total
**(ii)** $\frac{1}{2} \times 0.8v^2 = 0.8g \times 2.5 - 0.6 \times 5.6$ | A1ft |
Speed at $B$ is 6.45 m s$^{-1}$ | A1 | 3 marks total | For using KE gain = PE loss – WD against resistance
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$A B C D$ is a semi-circular cross-section, in a vertical plane, of the inner surface of half a hollow cylinder of radius 2.5 m which is fixed with its axis horizontal. $A D$ is horizontal, $B$ is the lowest point of the cross-section and $C$ is at a height of 1.8 m above the level of $B$ (see diagram). A particle $P$ of mass 0.8 kg is released from rest at $A$ and comes to instantaneous rest at $C$.\\
(i) Find the work done on $P$ by the resistance to motion while $P$ travels from $A$ to $C$.
The work done on $P$ by the resistance to motion while $P$ travels from $A$ to $B$ is 0.6 times the work done while $P$ travels from $A$ to $C$.\\
(ii) Find the speed of $P$ when it passes through $B$.
\hfill \mbox{\textit{CAIE M1 2012 Q1 [5]}}