| Exam Board | OCR |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2014 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Motion on a slope |
| Type | Particle on slope with pulley |
| Difficulty | Standard +0.3 This is a standard M1 pulley-on-slope question requiring systematic application of SUVAT equations, Newton's second law, and friction calculations. While it has multiple parts and requires careful bookkeeping of which particle is where, all techniques are routine for M1 students and no novel insight is needed—slightly easier than average due to the structured guidance through each part. |
| Spec | 3.03k Connected particles: pulleys and equilibrium3.03v Motion on rough surface: including inclined planes6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(s = 0.6 \times 2 + 0.9 \times 2^2/2\) | M1 | Uses \(s = ut + at^2/2\), \(u \neq 0\), \(a \neq g\) or \(g\text{CorS}30\) |
| \(s = 3\) | A1 | |
| \(AB = 6 \text{ m}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(V_M = 0.6 + 0.9 \times 2\) OR \(V_M^2 = 0.6^2 + 2 \times 0.9 \times 3\) | B1 | \(2.4\) / \(5.76\); Award if found in (i) and used in (ii) |
| \(a = g\sin 30\) | B1 | \(4.9\) |
| \(V_B^2 = 2.4^2 + 2(9.8\sin 30) \times 3\) | M1 | Uses \(v^2 = u^2 + 2as\), \(u \neq 0\) or \(0.6\), \(a \neq g\) or \(0.9\), \(s \neq AB(i)\); If \(AB(i) = 3\), allow its use for final M1A1 |
| \(V_B = 5.93 \text{ ms}^{-1}\) | A1 | Accept \(5.9\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.3 \times 0.9 = 0.3g\sin 30 - T\) | M1 | N2L, \(0.3 \times 0.9 = +/-(0.3g\text{CorS}30 - T)\); \(a = 0.9\) essential, \(m = 0.3\) but if \(0.4\) used in (iii) AND \(0.3\) used in (iv), treat as a single mis-read |
| A1 | ||
| \(T = 1.2 \text{ N}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.4 \times 0.9 = 0.4g\sin 30 + 1.2 - Fr\) | M1* | N2L, 3 forces inc \(+/-(0.4g\text{CorS}30 + T)\), ft \(cv(T)\) in (iii); \(a = 0.9\) or value used in (iii), \(m = 0.4\) but if \(0.4\) used in (iii) AND \(0.3\) used in (iv), treat as a single mis-read |
| A1ft | ||
| \(Fr = 2.8\) | A1 | May be shown by mu calculation |
| \(R = 0.4g\cos 30\) | B1 | May be implied, \(3.39(48...)\) |
| \(\mu = 2.8/3.39\) | D*M1 | \(2.8 = 3.39(48)\ \mu\), both forces positive; Awarded only if M1 for N2L equation |
| \(\mu = 0.825\) | A1 | Accept \(0.82\), not \(0.83\) or \(0.826\) |
# Question 7:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $s = 0.6 \times 2 + 0.9 \times 2^2/2$ | M1 | Uses $s = ut + at^2/2$, $u \neq 0$, $a \neq g$ or $g\text{CorS}30$ |
| $s = 3$ | A1 | |
| $AB = 6 \text{ m}$ | A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $V_M = 0.6 + 0.9 \times 2$ OR $V_M^2 = 0.6^2 + 2 \times 0.9 \times 3$ | B1 | $2.4$ / $5.76$; Award if found in (i) and used in (ii) |
| $a = g\sin 30$ | B1 | $4.9$ |
| $V_B^2 = 2.4^2 + 2(9.8\sin 30) \times 3$ | M1 | Uses $v^2 = u^2 + 2as$, $u \neq 0$ or $0.6$, $a \neq g$ or $0.9$, $s \neq AB(i)$; If $AB(i) = 3$, allow its use for final M1A1 |
| $V_B = 5.93 \text{ ms}^{-1}$ | A1 | Accept $5.9$ |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.3 \times 0.9 = 0.3g\sin 30 - T$ | M1 | N2L, $0.3 \times 0.9 = +/-(0.3g\text{CorS}30 - T)$; $a = 0.9$ essential, $m = 0.3$ but if $0.4$ used in (iii) AND $0.3$ used in (iv), treat as a single mis-read |
| | A1 | |
| $T = 1.2 \text{ N}$ | A1 | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.4 \times 0.9 = 0.4g\sin 30 + 1.2 - Fr$ | M1* | N2L, 3 forces inc $+/-(0.4g\text{CorS}30 + T)$, ft $cv(T)$ in (iii); $a = 0.9$ or value used in (iii), $m = 0.4$ but if $0.4$ used in (iii) AND $0.3$ used in (iv), treat as a single mis-read |
| | A1ft | |
| $Fr = 2.8$ | A1 | May be shown by mu calculation |
| $R = 0.4g\cos 30$ | B1 | May be implied, $3.39(48...)$ |
| $\mu = 2.8/3.39$ | D*M1 | $2.8 = 3.39(48)\ \mu$, both forces positive; Awarded only if M1 for N2L equation |
| $\mu = 0.825$ | A1 | Accept $0.82$, not $0.83$ or $0.826$ |
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\includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-5_510_1091_269_479}\\
$A$ and $B$ are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at $30 ^ { \circ }$ to the horizontal. $M$ is the mid-point of $A B$. Two particles $P$ and $Q$, joined by a taut light inextensible string, are placed on the plane at $A$ and $M$ respectively. The particles are simultaneously projected with speed $0.6 \mathrm {~ms} ^ { - 1 }$ down the line of greatest slope (see diagram). The particles move down the plane with acceleration $0.9 \mathrm {~ms} ^ { - 2 }$. At the instant 2 s after projection, $P$ is at $M$ and $Q$ is at $B$. The particle $Q$ subsequently remains at rest at $B$.\\
(i) Find the distance $A B$.
The plane is rough between $A$ and $M$, but smooth between $M$ and $B$.\\
(ii) Calculate the speed of $P$ when it reaches $B$.\\
$P$ has mass 0.4 kg and $Q$ has mass 0.3 kg .\\
(iii) By considering the motion of $Q$, calculate the tension in the string while both particles are moving down the plane.\\
(iv) Calculate the coefficient of friction between $P$ and the plane between $A$ and $M$.
\section*{END OF QUESTION PAPER}
\hfill \mbox{\textit{OCR M1 2014 Q7 [16]}}