| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2017 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Pulley systems |
| Type | Particle on rough incline connected to particle on horizontal surface or other incline |
| Difficulty | Standard +0.3 This is a standard two-particle pulley system with connected motion. Part (i) involves straightforward application of F=ma to both particles with smooth surfaces (5 marks for routine setup and algebra). Part (ii) adds friction forces requiring calculation of normal reactions and using kinematic equations, but follows a predictable method. The given sin θ = 3/5 simplifies calculations. While requiring multiple steps and careful bookkeeping across 14 marks total, this represents a typical M1 pulley question with no novel insights required—slightly easier than average due to its methodical nature. |
| Spec | 3.02c Interpret kinematic graphs: gradient and area3.03k Connected particles: pulleys and equilibrium3.03o Advanced connected particles: and pulleys3.03v Motion on rough surface: including inclined planes |
| Answer | Marks |
|---|---|
| 6(i) | A [T = 0.3a] |
| Answer | Marks | Guidance |
|---|---|---|
| System [1.5g sin θ = 1.8a] | M1 | Apply Newton’s second law to A or to B or to the system |
| A1 | Any two correct equations | |
| M1 | Solve 2 simultaneous equations for a and/or T or use the |
| Answer | Marks | Guidance |
|---|---|---|
| a = 9/1.8 = 5 ms–2 | A1 | |
| T = 1.5 N | A1 | |
| Total: | 5 | |
| Question | Answer | Marks |
| Answer | Marks | Guidance |
|---|---|---|
| 6(ii) | [5 = 3a] | M1 |
| a = 5/3 = 1.67 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| A B | B1 | For either reaction |
| Answer | Marks | Guidance |
|---|---|---|
| A B | M1 | Use F = µR for either term |
| Answer | Marks | Guidance |
|---|---|---|
| A B | (M1 | Apply Newton’s second law to A or to B or to the system |
| A2/1/0 | A1 Correct equation for A or B |
| Answer | Marks | Guidance |
|---|---|---|
| [9 – 15µ = 3] | M1 | Solve for µ from equations with correct number of terms |
| µ = 0.4 = 2/5 | A1) |
| Answer | Marks | Guidance |
|---|---|---|
| s = ½ (5/3) × 32 = 7.5 | (B1 | Find distance travelled in 3 secs |
| Answer | Marks | Guidance |
|---|---|---|
| 1.5 × 10 × 7.5 × (3/5) = 67.5 | B1 | |
| KE gain = ½ (1.8) × 52 = 22.5 | B1 | |
| [67.5 = 22.5 + 3µ × 7.5 + 12µ × 7.5] | M1 | Use Work/Energy equation |
| µ = 2/5 = 0.4 | A1) | |
| Total: | 9 |
Question 6:
--- 6(i) ---
6(i) | A [T = 0.3a]
B [1.5g sin θ – T = 1.5a]
System [1.5g sin θ = 1.8a] | M1 | Apply Newton’s second law to A or to B or to the system
A1 | Any two correct equations
M1 | Solve 2 simultaneous equations for a and/or T or use the
system equation.
a = 9/1.8 = 5 ms–2 | A1
T = 1.5 N | A1
Total: | 5
Question | Answer | Marks | Guidance
--- 6(ii) ---
6(ii) | [5 = 3a] | M1 | v = u + at used with t = 3, u = 0, v = 5
a = 5/3 = 1.67 | A1
R = 3 R = 15 cos 36.9 = 12
A B | B1 | For either reaction
[F = 3µ F = 12µ]
A B | M1 | Use F = µR for either term
EITHER:
A [T – F = 0.3a]
A
B [15 sin 36.9 – T – F = 1.5a]
B
System equation is
[1.5g sin 36.9 – F – F = 1.8a]
A B | (M1 | Apply Newton’s second law to A or to B or to the system
A2/1/0 | A1 Correct equation for A or B
A2 Correct equations for A and B
OR A2 Correct system equation
[9 – 15µ = 3] | M1 | Solve for µ from equations with correct number of terms
µ = 0.4 = 2/5 | A1)
OR:
s = ½ (5/3) × 32 = 7.5 | (B1 | Find distance travelled in 3 secs
PE loss =
1.5 × 10 × 7.5 × (3/5) = 67.5 | B1
KE gain = ½ (1.8) × 52 = 22.5 | B1
[67.5 = 22.5 + 3µ × 7.5 + 12µ × 7.5] | M1 | Use Work/Energy equation
µ = 2/5 = 0.4 | A1)
Total: | 9\includegraphics{figure_6}
The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of $\theta°$ to the horizontal, where $\sin \theta = \frac{3}{5}$. A particle $A$ of mass $0.3$ kg rests on the horizontal surface and is attached to one end of a light inextensible string. The string passes over a small smooth pulley $P$ fixed at the edge of the block. The other end of the string is attached to a particle $B$ of mass $1.5$ kg which rests on the sloping surface of the block. The system is released from rest with the string taut.
\begin{enumerate}[label=(\roman*)]
\item Given that the block is smooth, find the acceleration of particle $A$ and the tension in the string.
[5]
\item It is given instead that the block is rough. The coefficient of friction between $A$ and the block is $\mu$ and the coefficient of friction between $B$ and the block is also $\mu$. In the first $3$ seconds of the motion, $A$ does not reach $P$ and $B$ does not reach the bottom of the sloping surface. The speed of the particles after $3$ s is $5$ m s$^{-1}$. Find the acceleration of particle $A$ and the value of $\mu$.
[9]
\end{enumerate}
\hfill \mbox{\textit{CAIE M1 2017 Q6 [14]}}