| Exam Board | OCR MEI |
|---|---|
| Module | Further Mechanics A AS (Further Mechanics A AS) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Folded lamina |
| Difficulty | Standard +0.8 This is a multi-part Further Maths mechanics question requiring: (a) standard 2D centre of mass for a triangle, (b) 3D centre of mass calculation with composite shapes after folding (requiring spatial visualization and careful coordinate tracking), and (c) equilibrium analysis checking if the centre of mass lies within the base of support. The 3D folding aspect and equilibrium condition elevate this above typical A-level questions, but the techniques are standard for Further Maths mechanics. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | (1(0+20+100),1(0+60+0)) |
| Answer | Marks | Guidance |
|---|---|---|
| 3 3 | M1 | 1.2 |
| Answer | Marks | Guidance |
|---|---|---|
| 40,20 | A1 | 1.1 |
| Answer | Marks |
|---|---|
| (b) | ( ) |
| Answer | Marks | Guidance |
|---|---|---|
| 5000x =3000×40+1000×70+1000×28 | M1 | 3.1b |
| Answer | Marks | Guidance |
|---|---|---|
| x =43.6 | A1 | 1.1 |
| 5000y =3000×20+1000×0+1000×54 | M1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 5000z | M1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| y =22.8, z =−10 | A1 | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| (c) | Line joining ( 20,60 ) to ( 60,0 ) has equation y=−3x+90. | |
| 2 | M1 | 3.1b |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | 1.1 | Finding one critical value |
| Answer | Marks | Guidance |
|---|---|---|
| ⇒ it will topple. | A1ft | 2.2a |
| Answer | Marks |
|---|---|
| 39.71 <y < 42.3 ⇒unstable | y must be |
Question 7:
7 | (a) | (1(0+20+100),1(0+60+0))
Centre of mass lies at
3 3 | M1 | 1.2 | One coordinate sufficient. May be
implied by one correct answer.
=( )
40,20 | A1 | 1.1
[2]
(b) | ( )
Let the COM lie at x,y,z .
5000x =3000×40+1000×70+1000×28 | M1 | 3.1b | Might be seen embedded in a
vector equation with y and z .
Award if all terms present and at
most one error in areas and at most
one error in x-coordinates 40 (ft),
28, 70
x =43.6 | A1 | 1.1 | AG
5000y =3000×20+1000×0+1000×54 | M1 | 1.1 | Allow one error in y-coordinates
20 (ft), 54, 0
=3000×0+1000×(−25 )+1000×(−25 )
5000z | M1 | 1.1 | Allow one error in z-coordinates
0, ˗25 (but -50 is M0)
y =22.8, z =−10 | A1 | 1.1 | cao
[5]
(c) | Line joining ( 20,60 ) to ( 60,0 ) has equation y=−3x+90.
2 | M1 | 3.1b | Or Line joining (36, 48) to (80, 0)
is y = ˗ (12/11)x + (960/11)
M1 | 1.1 | Finding one critical value
(24.6 or 2184/55 ≈ 39.71)
3
y = 22.8 < 24.6, so COM lies outside the base region
− (43.6)+90 = 24.6
2
⇒ it will topple. | A1ft | 2.2a | Follow through their value of y
provided 0 < y < 42.3
0 <y < 24.6 ⇒unstable
24.6 <y < 39.71 ⇒stable
39.71 <y < 42.3 ⇒unstable | y must be
compared with
the correct
relevant
critical
value(s)
[3]
PMT
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recorded or monitored7 The vertices of a uniform triangular lamina, which is in the $x - y$ plane, are at the origin and the points $( 20,60 )$ and $( 100,0 )$.
\begin{enumerate}[label=(\alph*)]
\item Determine the coordinates of the lamina's centre of mass.
Fig. 7.1 shows a uniform lamina consisting of a triangular section and two identical rectangular sections. The coordinates of some of the vertices of the lamina are given in Fig. 7.1.
The rectangular sections are then folded at right-angles to the triangular section, to give the rigid three-dimensional object illustrated in Fig. 7.2. Two of the edges, $E _ { 1 }$ and $E _ { 2 }$, are marked on both figures.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-7_933_739_799_164}
\captionsetup{labelformat=empty}
\caption{Fig. 7.1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-7_924_725_808_1133}
\captionsetup{labelformat=empty}
\caption{Fig. 7.2}
\end{center}
\end{figure}
\item Show that the $x$-coordinate of the centre of mass of the folded object is 43.6, and determine the $y$ - and $z$-coordinates.
\item Determine whether it is possible for the folded object to rest in equilibrium with edges $E _ { 1 }$ and $E _ { 2 }$ in contact with a horizontal surface.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2021 Q7 [10]}}