| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2018 |
| Session | June |
| Marks | 12 |
| Topic | Pulley systems |
| Type | String breaks during motion |
| Difficulty | Challenging +1.8 This is a challenging multi-stage mechanics problem requiring: (1) resolving forces on an inclined plane with friction, (2) applying Newton's second law to a pulley system, (3) analyzing motion after string becomes slack when A hits ground, (4) determining when B comes to rest and using kinematics across two phases. The conceptual demand of tracking the system through the transition when A hits the ground, plus the multi-step calculation with friction and inclined plane components, places this well above average difficulty but within reach of strong A-level students. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03o Advanced connected particles: and pulleys3.03v Motion on rough surface: including inclined planes |
**Question 12(i)**
$Fr = 0.4 \times 2g\cos20$ **B1** (7.517540966)
Attempt to use Newton's second law on either mass or the system **M1**
$3g - T = 3a$ **A1** System equation: $3g - 2g\sin20 - Fr = 5a$
$T - 2g\sin20 - Fr = 2a$ **A1**
Solve for $a$ or $T$ **M1**
$a = 3.13\ \text{ms}^{-1}$ **A1**
$T = 20.6\ \text{N}$ **B1**
**Question 12(ii)**
Use $v^2 = 2 \times 3.13 \times 1.2$ to find $v$ **M1** Using *their* 3.13
$(2)a = (2)g\sin20 + 0.4 \times (2)g\cos20$ **M1** Using *their* $Fr$; $a = 7.178971916$
Use $v = u + at$ to find either time **M1** or use $s = ut + \frac{1}{2}at^2$ on $A$
Get either $0.876$ or $0.382$ **A1**
Get $(0.876 + 0.382) = 1.26\ \text{s}$ **A1**12\\
\includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-5_429_873_264_635}
The diagram shows a block $B$ of mass 2 kg and a particle $A$ of mass 3 kg attached to opposite ends of a light inextensible string. The block is held at rest on a rough plane inclined at $20 ^ { \circ }$ to the horizontal, and the coefficient of friction between the block and the plane is 0.4 . The string passes over a small smooth pulley $C$ at the edge of the plane and $A$ hangs in equilibrium 1.2 m above horizontal ground. The part of the string between $B$ and $C$ is parallel to a line of greatest slope of the plane. $B$ is released and begins to move up the plane.\\
(i) Show that the acceleration of $A$ is $3.13 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, correct to 3 significant figures, and find the tension in the string.\\
(ii) When $A$ reaches the ground it remains there. Given that $B$ does not reach $C$ in the subsequent motion, find the total time that $B$ is moving up the plane.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2018 Q12 [12]}}