| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Forces, equilibrium and resultants |
| Type | Equilibrium of particle under coplanar forces |
| Difficulty | Standard +0.3 This is a standard 3-force equilibrium problem requiring resolution of forces and geometric reasoning with the sphere. Part (i) involves setting up force equations with weight, normal reactions from wall and plane, using the geometry to find angles. Part (ii) requires recognizing that equilibrium breaks down when one contact force becomes zero. While it requires careful geometric setup and algebraic manipulation, it follows a well-established method for sphere-on-incline problems that students practice extensively in M2, making it slightly easier than average. |
| Spec | 3.04b Equilibrium: zero resultant moment and force6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks |
|---|---|
| 5 (i) | v2 = (15sin30) 2 + 2 g × 20 |
| Answer | Marks |
|---|---|
| (= tan 21.36/(15cos30) = 58.7 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| A1 | [4] | v = 21.3600.. |
| Answer | Marks |
|---|---|
| OR | –20 = (15sin30)t – gt2 /2 |
| Answer | Marks |
|---|---|
| OP = 42.5 m | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| A1 | [4] | M1 maybe gained in (i) |
| Answer | Marks |
|---|---|
| 42.491.. | 8 |
Question 5:
--- 5 (i) ---
5 (i) | v2 = (15sin30) 2 + 2 g × 20
v
V2 2 2
= (15cos30) + (15sin30) +
2 g × 20
V = 25 ms −1
θ −1 o
(= tan 21.36/(15cos30) = 58.7 | M1
M1
A1
A1 | [4] | v = 21.3600..
v
V or θ from components, V by energy
58.69..
(ii)
OR | –20 = (15sin30)t – gt2 /2
2t2 – 3t – 8 = 0 ⇒t(= 2.866..) = 2.89
t = (15sin30)/10 + (25sin58.7)/10
t = 2.89
OP2 =20 2 + (15cos30 × 2.886) 2
OP = 42.5 m | M1
A1
M1
A1
M1
A1 | [4] | M1 maybe gained in (i)
A1 maybe gained in (i)
Separating rise and fall times
42.491.. | 8A smooth sphere of mass $M$ and radius $a$ rests in contact with a smooth vertical wall and a smooth inclined plane. The plane makes an angle $\alpha$ with the horizontal.
\begin{enumerate}[label=(\roman*)]
\item Find the magnitude of each of the contact forces acting on the sphere.
\item Find the range of values of $\alpha$ for which this equilibrium is possible.
\end{enumerate}
[8]
\hfill \mbox{\textit{CAIE M2 2013 Q5 [8]}}