| Exam Board | OCR MEI |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Conical or hemispherical shell composite |
| Difficulty | Standard +0.8 This is a substantial multi-part 3D centre of mass problem requiring systematic calculation of composite bodies with changing configurations, followed by moments analysis on an inclined plane. While the individual techniques (finding centres of mass of rectangular faces, combining them proportionally) are standard M2 content, the question demands careful spatial reasoning across multiple configurations, coordinate geometry in 3D, and precise moment calculations. The extended nature (18 marks total) and need to track changing geometry elevate this above a routine exercise, though it remains methodical rather than requiring novel insight. |
| Spec | 3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids |
| Answer | Marks |
|---|---|
| R | N |
Question 3:
3
2
YN
XN A B C
fixed
LN
RN
D E
60∞
Fig. 2
Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods AB, BC, AD,
BD, BE, CE and DE. [The triangles ABD, BDE and BCE are all equilateral.]
The rods AB, BC and DE are horizontal.
The rods are freely pin-jointed to each other at A, B, C, D and E.
The pin-joint at Ais also fixed to an inclined plane. The plane is smooth and parallel to the rod AD.
The pin-joint at D rests on this plane.
The following external forces act on the framework: a vertical load of LN at C; the normal reaction
force R N of the plane on the framework at D; the horizontal and vertical forces X N and Y N,
respectively, acting at A.
(i) Write down equations for the horizontal and vertical equilibrium of the framework. [3]
(ii) By considering moments, find the relationship between R and L. Hence show that X = 3L
and Y (cid:2) 0. [4]
(iii) Draw a diagram showing all the forces acting on the pin-joints, including the forces internal
to the rods. [2]
(iv) Show that the internal force in the rod AD is zero. [2]
(v) Find the forces internal to AB, CE and BC in terms of L and state whether each is a tension
or a thrust (compression). [You may leave your answers in surd form.] [7]
(vi) Without calculating their values in terms of L, show that the forces internal to the rods BD and
BE have equal magnitude but one is a tension and the other a thrust. [2]
[Turn over
© OCR 2007 4762/01 Jan 07
R | NA box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure.
\includegraphics{figure_3.1}
\begin{enumerate}[label=(\roman*)]
\item The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. [1]
\end{enumerate}
The base OABC is added to the vertical faces.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Write down the $x$- and $y$-coordinates of the centre of mass of the box now. Show that the $z$-coordinate is now 1.875. [5]
\end{enumerate}
The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Show that the coordinates of the centre of mass of the box in this situation are $(10, 2.4, 2.1)$. [6]
\end{enumerate}
[This question is continued on the facing page.]
The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at $30°$ to the horizontal, as shown in Fig. 3.2.
\includegraphics{figure_3.2}
The weight of the box is 40 N. A force $P$ N acts parallel to the plane and is applied to the mid-point of FG at $90°$ to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm. [4]
\item Calculate the value of $P$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI M2 2007 Q3 [18]}}