OCR MEI Further Mechanics A AS 2020 November — Question 6 10 marks

Exam BoardOCR MEI
ModuleFurther Mechanics A AS (Further Mechanics A AS)
Year2020
SessionNovember
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCentre of Mass 1
TypeToppling on inclined plane
DifficultyStandard +0.8 This is a multi-step 3D centre of mass problem requiring decomposition of a trapezoidal prism, calculation of coordinates in 3D space, and then a toppling analysis using geometric reasoning about whether the vertical line through the centre of mass falls within the base of support. It combines several techniques (composite bodies, 3D coordinates, stability conditions) and requires careful geometric visualization, making it moderately challenging but still within standard Further Maths scope.
Spec6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces
6 Fig. 6.1 shows a solid uniform prism OABCDEFG . The \(\mathrm { Ox } , \mathrm { Oy }\) and Oz axes are also shown. The cross-section of the prism is a trapezium. Fig. 6.2 shows the face OABC . The dimensions shown in the figures are in centimetres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-6_528_672_571_274} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-6_524_538_571_1242} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
\end{figure} The centre of mass of the prism has coordinates \(( \bar { x } , \bar { y } , \bar { z } )\).
  1. Determine the values of \(\bar { x } , \bar { y }\) and \(\bar { z }\).
  2. By considering triangle PBA, where P has coordinates ( \(\bar { x } , 0 , \bar { z }\) ), determine whether it is possible for the prism to rest with the face ABEF in contact with a horizontal plane without toppling.
Question 6:
AnswerMarks Guidance
6(a) y =4.5
Area of OABC =72B1 1.1
Coordinates of centre of mass of triangle =( 11,2 )B1 1.1
72x =9×6×4.5+ 1×6×6×11
AnswerMarks Guidance
2M1 3.3
of mass – correct number of terms and
dimensionally consistent
AnswerMarks Guidance
x =6.125A1 1.1
72z =9×6×3+ 1×6×6×2
AnswerMarks Guidance
2M1 3.3
of mass – correct number of terms and
dimensionally consistent
AnswerMarks Guidance
z =2.75A1 1.1
[7]
AnswerMarks
(b)Method 1
PB2 +BA2 =(9−6.125)2 +(6−2.75)2 +62 +62 = 5813
AnswerMarks Guidance
64M1 3.1b
PA2 =(15−6.125)2 +(0−2.75)2 = 5525 <PB2 +BA2
AnswerMarks Guidance
64M1 1.1
∴∠PBA<90° so yes, the prism will stay in that position.A1 3.2a
Method 2[3]
6.125−9 15−9
   ⋅ 
BP⋅BA  2.75−6   0−6 
cos(∠PBA)= =
AnswerMarks Guidance
BP ⋅ BA 2.8752 +3.252 62 +62M1 Or direct use of Cosine Rule may be used
2.25
cos(∠PBA)=
AnswerMarks
18.8 72M1
∠PBA≈86.5°<90° so yes, the prism will stay in that
AnswerMarks
position.A1 
Question 6:
6 | (a) | y =4.5 | B1 | 1.1
Area of OABC =72 | B1 | 1.1
Coordinates of centre of mass of triangle =( 11,2 ) | B1 | 1.1 | s.o.i.
72x =9×6×4.5+ 1×6×6×11
2 | M1 | 3.3 | Correct method for finding x-coordinate of centre
of mass – correct number of terms and
dimensionally consistent
x =6.125 | A1 | 1.1
72z =9×6×3+ 1×6×6×2
2 | M1 | 3.3 | Correct method for finding z-coordinate of centre
of mass – correct number of terms and
dimensionally consistent
z =2.75 | A1 | 1.1
[7]
(b) | Method 1
PB2 +BA2 =(9−6.125)2 +(6−2.75)2 +62 +62 = 5813
64 | M1 | 3.1b
PA2 =(15−6.125)2 +(0−2.75)2 = 5525 <PB2 +BA2
64 | M1 | 1.1
∴∠PBA<90° so yes, the prism will stay in that position. | A1 | 3.2a
Method 2 | [3]
6.125−9 15−9
   ⋅ 
BP⋅BA  2.75−6   0−6 
cos(∠PBA)= =
BP ⋅ BA 2.8752 +3.252 62 +62 | M1 | Or direct use of Cosine Rule may be used
2.25
cos(∠PBA)=
18.8 72 | M1
∠PBA≈86.5°<90° so yes, the prism will stay in that
position. | A1
6 Fig. 6.1 shows a solid uniform prism OABCDEFG . The $\mathrm { Ox } , \mathrm { Oy }$ and Oz axes are also shown. The cross-section of the prism is a trapezium.

Fig. 6.2 shows the face OABC .

The dimensions shown in the figures are in centimetres.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-6_528_672_571_274}
\captionsetup{labelformat=empty}
\caption{Fig. 6.1}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{b20e2254-955e-466c-8161-9614d8ccdba0-6_524_538_571_1242}
\captionsetup{labelformat=empty}
\caption{Fig. 6.2}
\end{center}
\end{figure}

The centre of mass of the prism has coordinates $( \bar { x } , \bar { y } , \bar { z } )$.
\begin{enumerate}[label=(\alph*)]
\item Determine the values of $\bar { x } , \bar { y }$ and $\bar { z }$.
\item By considering triangle PBA, where P has coordinates ( $\bar { x } , 0 , \bar { z }$ ), determine whether it is possible for the prism to rest with the face ABEF in contact with a horizontal plane without toppling.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2020 Q6 [10]}}
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