OCR MEI Further Mechanics A AS 2019 June — Question 4 9 marks

Exam BoardOCR MEI
ModuleFurther Mechanics A AS (Further Mechanics A AS)
Year2019
SessionJune
Marks9
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Mark schemeDownload PDF ↗
TopicCentre of Mass 1
TypeParticles at coordinate positions
DifficultyStandard +0.3 This is a straightforward 3D centre of mass problem requiring standard formulas for composite bodies (rectangular lamina and rod). Part (a) involves routine weighted average calculations with clearly given masses and dimensions. Part (b) applies equilibrium conditions to find an angle, which is a standard application. The 3D aspect and trigonometry add minor complexity, but the problem follows a predictable template with no novel insight required.
Spec6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces
4 A shovel consists of a blade and handle, as shown in Fig. 4.1 and Fig. 4.2. The dimensions shown in the figures are in metres.
The blade is modelled as a uniform rectangular lamina ABCD lying in the Oxy plane, where O is the mid-point of AB . The handle is modelled as a thin uniform rod EF . The handle lies in the Oyz plane, and makes an angle \(\alpha\) with \(\mathrm { O } y\), where \(\sin \alpha = \frac { 7 } { 25 }\). The rod and lamina are rigidly attached at E, the mid-point of CD.
The blade of the shovel has mass 1.25 kg and the handle of the shovel has mass 0.5 kg . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-3_746_671_1217_246} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-3_664_766_1226_1064} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
\end{figure}
  1. Find,
    1. the \(y\)-coordinate of the centre of mass of the shovel,
    2. the \(z\)-coordinate of the centre of mass of the shovel. The shovel is freely suspended from O and hangs in equilibrium.
  2. Calculate the angle that OE makes with the vertical.
Question 4:
AnswerMarks Guidance
4(a) (i)
0.125  1.25 
0.5  0.250.6cos
AnswerMarks
y0.325M1
B1
A1
A1
AnswerMarks
A13.1b
2.1
1.1
1.1
AnswerMarks
2.2aTable of values idea soi – both
blade and handle used
Correct LHS
Correct first term on RHS
Correct second terms on RHS
AnswerMarks
2 s.f. or better0.15625
0.25 + 0.576 = 0.826
2277
0.3252857143… or
7000
[5]
AnswerMarks Guidance
(a)(ii)  1.250.5 z 0.5  0.6sin
wrong mass / length on
right 0.084
AnswerMarks Guidance
z0.048A1
[2]1.1 cao
(b)0.048
tan
AnswerMarks Guidance
0.325...M1 3.4
and (b) – allow reciprocal
AnswerMarks Guidance
8.4°A1 1.1
[2]
Question 4:
4 | (a) | (i) |  1.250.5 y
0.125  1.25 
0.5  0.250.6cos
y0.325 | M1
B1
A1
A1
A1 | 3.1b
2.1
1.1
1.1
2.2a | Table of values idea soi – both
blade and handle used
Correct LHS
Correct first term on RHS
Correct second terms on RHS
2 s.f. or better | 0.15625
0.25 + 0.576 = 0.826
2277
0.3252857143… or
7000
[5]
(a) | (ii) |  1.250.5 z 0.5  0.6sin | M1 | 1.1 | Table of values idea soi | Allow sin/cos confusion;
wrong mass / length on
right 0.084
z0.048 | A1
[2] | 1.1 | cao
(b) | 0.048
tan
0.325... | M1 | 3.4 | Use of tan with their answers to (a)
and (b) – allow reciprocal
8.4° | A1 | 1.1 | Cao; www | 8.4014… − 8.394136…
[2]
4 A shovel consists of a blade and handle, as shown in Fig. 4.1 and Fig. 4.2. The dimensions shown in the figures are in metres.\\
The blade is modelled as a uniform rectangular lamina ABCD lying in the Oxy plane, where O is the mid-point of AB . The handle is modelled as a thin uniform rod EF . The handle lies in the Oyz plane, and makes an angle $\alpha$ with $\mathrm { O } y$, where $\sin \alpha = \frac { 7 } { 25 }$. The rod and lamina are rigidly attached at E, the mid-point of CD.\\
The blade of the shovel has mass 1.25 kg and the handle of the shovel has mass 0.5 kg .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-3_746_671_1217_246}
\captionsetup{labelformat=empty}
\caption{Fig. 4.1}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-3_664_766_1226_1064}
\captionsetup{labelformat=empty}
\caption{Fig. 4.2}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Find,
\begin{enumerate}[label=(\roman*)]
\item the $y$-coordinate of the centre of mass of the shovel,
\item the $z$-coordinate of the centre of mass of the shovel.

The shovel is freely suspended from O and hangs in equilibrium.
\end{enumerate}\item Calculate the angle that OE makes with the vertical.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2019 Q4 [9]}}
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