| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2007 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton's laws and connected particles |
| Type | Block on rough horizontal surface – equilibrium (finding friction, normal reaction, or coefficient of friction) |
| Difficulty | Standard +0.3 This is a standard two-part equilibrium problem requiring resolution of forces in horizontal and vertical directions. Part (i) involves basic force resolution on a smooth surface (3-4 equations), while part (ii) adds friction with the coefficient given by μ = F/R. The setup is straightforward with clearly defined angles and forces, requiring only systematic application of ΣFₓ = 0 and ΣFᵧ = 0 with no conceptual challenges beyond standard M1 techniques. |
| Spec | 3.03m Equilibrium: sum of resolved forces = 03.03n Equilibrium in 2D: particle under forces3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(T\cos60° = 75\cos30° \Rightarrow T = 130\) | B1 | Accept \(75\sqrt{3}\) |
| M1 | For resolving forces vertically (4 terms) | |
| \(T\sin60° + 75\sin30° + R = 20g\) | A1ft | ft consistent sin/cos mix |
| \([130\sin60° + 75\sin30° + R = 200]\) | M1 | For substituting for \(T\) and solving for \(R\) |
| Magnitude is 50 N | A1 | Accept 49.9 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| M1 | For resolving forces horizontally | |
| \(T\cos60° + 25 = 75\cos30°\), \((T = 79.9)\) | A1ft | ft consistent sin/cos mix (\(T = 14.4\)) |
| \([79.9\sin60° + 75\sin30° + R = 200]\) | M1 | For resolving forces vertically (4 terms) and substituting for \(T\) |
| \(R = 93.3\) | A1 | May be implied by final answer |
| \([\mu = 25/93.3]\) | M1 | For using \(\mu = 25/R\) |
| Coefficient is \(0.268\ (= 2 - \sqrt{3})\) | A1ft | ft for \(\mu\) = value obtained from 25/candidate's \(R\), including her/his answer in (i) but excluding \(R = 20g\) |
## Question 7:
### Part (i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $T\cos60° = 75\cos30° \Rightarrow T = 130$ | B1 | Accept $75\sqrt{3}$ |
| | M1 | For resolving forces vertically (4 terms) |
| $T\sin60° + 75\sin30° + R = 20g$ | A1ft | ft consistent sin/cos mix |
| $[130\sin60° + 75\sin30° + R = 200]$ | M1 | For substituting for $T$ and solving for $R$ |
| Magnitude is 50 N | A1 | Accept 49.9 |
### Part (ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| | M1 | For resolving forces horizontally |
| $T\cos60° + 25 = 75\cos30°$, $(T = 79.9)$ | A1ft | ft consistent sin/cos mix ($T = 14.4$) |
| $[79.9\sin60° + 75\sin30° + R = 200]$ | M1 | For resolving forces vertically (4 terms) and substituting for $T$ |
| $R = 93.3$ | A1 | May be implied by final answer |
| $[\mu = 25/93.3]$ | M1 | For using $\mu = 25/R$ |
| Coefficient is $0.268\ (= 2 - \sqrt{3})$ | A1ft | ft for $\mu$ = value obtained from 25/candidate's $R$, including her/his answer in (i) but excluding $R = 20g$ |7\\
\includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_414_865_1512_641}
Two light strings are attached to a block of mass 20 kg . The block is in equilibrium on a horizontal surface $A B$ with the strings taut. The strings make angles of $60 ^ { \circ }$ and $30 ^ { \circ }$ with the horizontal, on either side of the block, and the tensions in the strings are $T \mathrm {~N}$ and 75 N respectively (see diagram).\\
(i) Given that the surface is smooth, find the value of $T$ and the magnitude of the contact force acting on the block.\\
(ii) It is given instead that the surface is rough and that the block is on the point of slipping. The frictional force on the block has magnitude 25 N and acts towards $A$. Find the coefficient of friction between the block and the surface.
\hfill \mbox{\textit{CAIE M1 2007 Q7 [11]}}