| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Topic | Pulley systems |
| Type | Heavier particle hits ground, lighter continues upward - vertical strings |
| Difficulty | Standard +0.3 This is a standard Atwood machine problem requiring Newton's second law for connected particles, followed by routine kinematics and impulse calculations. While it involves multiple parts and mechanics concepts (forces, acceleration, impulse), each step follows directly from standard A-level mechanics techniques with no novel problem-solving required. The multi-part structure and mechanics context place it slightly above average difficulty. |
| Spec | 3.03c Newton's second law: F=ma one dimension3.03k Connected particles: pulleys and equilibrium6.03e Impulse: by a force6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) N2L & Resolve vertically for either particle | M1 A1 | Accept use of \(g = 9.8\) throughout. |
| \(0.3a = 0.3g - T\) | ||
| \(0.2a = T - 0.2g\) | ||
| \(\therefore a = 0.1g = 1\) | M1 A1 | Eliminate either \(T\) or \(a\). Correct value for one. c.a.o. |
| \(\therefore T = 0.3 \times 10 - 0.3 \times 2 = 2.4\) N | A1 | Correct value for the other. c.a.o. [6] |
| ALTERNATIVE: | ||
| N2L for whole system | M1 A1 | Allow 1 error. All correct. c.a.o. |
| \((0.3 + 0.2)a = 0.3g - 0.2g\) | ||
| \(\therefore a = 2\) ms\(^{-2}\) | ||
| N2L & Resolve vertically for either particle | M1 A1 | All correct. ft c's \(a\). |
| \(0.3a = 0.3g - T\) or \(0.2a = T - 0.2g\) | ||
| \(\therefore T = 2.4\) N | A1 | c.a.o. |
| (ii) \(v^2 = 0^2 + 2 \times 2 \times 2.25 = 9\) | M1 A1 | Use of appropriate 'suvat' equation. ft c's \(a\). [2] |
| \(\therefore v = 3\) ms\(^{-1}\) | ||
| (iii) \(I = (0.3 \times 3) - (0)\) | M1 A1 | Use of Impulse = change in momentum. ft c's \(v\), including units. Allow –0.9 and/or kgms\(^{-1}\). |
| \(= 0.9\) Ns | ||
| (iv) \(0.9 = P \times 0.005\) | M1 A1 | Use of Impulse = force × time, o.e. ft c's \(I\). Allow –180. [2] |
| \(\therefore P = 180\) N |
**(i)** N2L & Resolve vertically for either particle | M1 A1 | Accept use of $g = 9.8$ throughout.
$0.3a = 0.3g - T$ |
$0.2a = T - 0.2g$ |
$\therefore a = 0.1g = 1$ | M1 A1 | Eliminate either $T$ or $a$. Correct value for one. c.a.o.
$\therefore T = 0.3 \times 10 - 0.3 \times 2 = 2.4$ N | A1 | Correct value for the other. c.a.o. [6]
ALTERNATIVE: |
N2L for whole system | M1 A1 | Allow 1 error. All correct. c.a.o.
$(0.3 + 0.2)a = 0.3g - 0.2g$ |
$\therefore a = 2$ ms$^{-2}$ |
N2L & Resolve vertically for either particle | M1 A1 | All correct. ft c's $a$.
$0.3a = 0.3g - T$ or $0.2a = T - 0.2g$ |
$\therefore T = 2.4$ N | A1 | c.a.o. |
**(ii)** $v^2 = 0^2 + 2 \times 2 \times 2.25 = 9$ | M1 A1 | Use of appropriate 'suvat' equation. ft c's $a$. [2]
$\therefore v = 3$ ms$^{-1}$ |
**(iii)** $I = (0.3 \times 3) - (0)$ | M1 A1 | Use of Impulse = change in momentum. ft c's $v$, including units. Allow –0.9 and/or kgms$^{-1}$.
$= 0.9$ Ns |
**(iv)** $0.9 = P \times 0.005$ | M1 A1 | Use of Impulse = force × time, o.e. ft c's $I$. Allow –180. [2]
$\therefore P = 180$ N |A light inextensible string passes over a smooth fixed pulley. Particles of mass 0.2 kg and 0.3 kg are attached to opposite ends of the string, so that the parts of the string not in contact with the pulley are vertical. The system is released from rest with the string taut.
\begin{enumerate}[label=(\roman*)]
\item Find the acceleration of the particles and the tension in the string. [6]
\end{enumerate}
When the heavier particle has fallen 2.25 m it hits the ground and is brought to rest (and the string goes slack).
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the speed with which it hits the ground. [2]
\item Find the magnitude of the impulse of the ground on the particle. [2]
\item If the impact between the particle and the ground lasts for 0.005 seconds, find the constant force that would be needed to bring the particle to rest. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2014 Q11 [12]}}