| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Mechanics (Further Paper 3 Mechanics) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Centre of mass of lamina by integration |
| Difficulty | Standard +0.8 This is a standard Further Maths mechanics question on centres of mass of laminas, requiring integration to find centroid coordinates and then applying equilibrium principles. While it involves multiple steps and careful calculation with the parabolic boundary, the techniques are well-practiced in Further Maths syllabi. The integration is straightforward (polynomials), and the equilibrium geometry is a direct application of standard principles. It's moderately challenging but follows a predictable template for this topic. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a) | Obtains correct mass or area, PI | 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Condone missing density | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone missing density | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| and for y , written as fractions. | 1.1b | A1 |
| Total | 4 | |
| Q | Marking Instructions | AO |
| Answer | Marks |
|---|---|
| 5(b) | Drawing a diagram showing the |
| Answer | Marks | Guidance |
|---|---|---|
| and the required angle, PI. | 3.4 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| angle | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| part a) | 1.1b | A1F |
| Total | 3 | |
| Question total | 7 | |
| Q | Marking Instructions | AO |
Question 5:
--- 5(a) ---
5(a) | Obtains correct mass or area, PI | 1.1b | B1 | Mass= 4( x2 +1 ) dx
0
76
=
3
x 76 = 4( x3 + x ) dx
3 0
76
x =72
3
54
x =
19
76 1 4
y = (x2 +1)2dx
3 2 0
76 3772
y =
3 30
943
y =
190
54 943
,
19 190
Sets up equation to find x using
appropriate integral
Condone missing density | 1.1a | M1
Sets up equation to find yusing
appropriate integral
Condone missing density | 1.1a | M1
Obtains correct exact value for x
and for y , written as fractions. | 1.1b | A1
Total | 4
Q | Marking Instructions | AO | Marks | Typical Solution
--- 5(b) ---
5(b) | Drawing a diagram showing the
CM and the point of suspension
and the required angle, PI. | 3.4 | M1 | 5 4
4 −
19 94
t a n =
3
1 7 −
1 9 0
5 = o
Uses their model to set up a
suitable trigonometric equation
using their coordinates to find the
angle | 1.1a | M1
Obtains the correct angle (FT from
part a) | 1.1b | A1F
Total | 3
Question total | 7
Q | Marking Instructions | AO | Marks | Typical SolutionA uniform lamina has the shape of the region enclosed by the curve $y = x^2 + 1$ and the lines $x = 0$, $x = 4$ and $y = 0$
The diagram below shows the lamina.
\includegraphics{figure_5}
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the centre of mass of the lamina, giving your answer in exact form.
[4 marks]
\item The lamina is suspended from the point where the curve intersects the line $x = 4$ and hangs in equilibrium.
Find the angle between the vertical and the longest straight edge of the lamina, giving your answer correct to the nearest degree.
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Mechanics 2021 Q5 [7]}}