Standard +0.3 This is a standard two-part centre of mass question requiring (i) composite body calculation by dividing the L-shape into rectangles, and (ii) taking moments about a pivot point for limiting equilibrium. Both are routine M2 techniques with straightforward arithmetic, making it slightly easier than average.
3
\(A B C D E F\) is the L -shaped cross-section of a uniform solid. This cross-section passes through the centre of mass of the solid and has dimensions as shown in Fig. 1.
Find the distance of the centre of mass of the solid from the edge \(A B\) of the cross-section.
The solid rests in equilibrium with the face containing the edge \(A F\) of the cross-section in contact with a horizontal table. The weight of the solid is \(W\) N. A horizontal force of magnitude \(P\) N is applied to the solid at the point \(B\), in the direction of \(B C\) (see Fig. 2). The table is sufficiently rough to prevent sliding.
Find \(P\) in terms of \(W\), given that the equilibrium of the solid is about to be broken.
For obtaining an equation in P and W by taking moments about F and using the idea that the normal component of the contact force has no moment about F (almost certainly implied in most cases). \(30P = 7.5W\) (moment about A) is M0
\((20 - 7.5)W = 30P\)
A1 ft
\(P = \frac{5}{12}W\ (= 0.417W)\)
A1
[3 marks]
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For obtaining an equation in $\bar{x}$ by taking moments (equation to contain all relevant terms) |
| $(300 + 100)\bar{x} = 300 \times 5 + 100 \times 15$ | A1 | Any correct equation in $\bar{x}$ |
| Distance is $7.5$ m | A1 | **[3 marks]** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For obtaining an equation in P and W by taking moments about F and using the idea that the normal component of the contact force has no moment about F (almost certainly implied in most cases). $30P = 7.5W$ (moment about A) is M0 |
| $(20 - 7.5)W = 30P$ | A1 ft | |
| $P = \frac{5}{12}W\ (= 0.417W)$ | A1 | **[3 marks]** |
---
3
\begin{tikzpicture}[>=Stealth, scale=0.08]
% Define coordinates of the L-shape
\coordinate (A) at (0,0);
\coordinate (B) at (0,40);
\coordinate (C) at (10,40);
\coordinate (D) at (10,10);
\coordinate (E) at (20,10);
\coordinate (F) at (20,0);
% Draw the L-shape
\draw[thick] (A) -- (B) -- (C) -- (D) -- (E) -- (F) -- cycle;
% Labels at vertices
\node[below left] at (A) {$A$};
\node[above left] at (B) {$B$};
\node[above right] at (C) {$C$};
\node[above right] at (D) {$D$};
\node[above right] at (E) {$E$};
\node[below right] at (F) {$F$};
% Dimension: 30 cm on the left (AB side, but showing full height)
\draw[<->] (-5,0) -- (-5,40);
\node[left] at (-5,20) {$30\,\mathrm{cm}$};
% Dimension: 10 cm on top (BC)
\draw[<->] (0,43) -- (10,43);
\node[above] at (5,43) {$10\,\mathrm{cm}$};
% Dimension: 10 cm on right (DE side height)
\draw[<->] (23,0) -- (23,10);
\node[right] at (23,5) {$10\,\mathrm{cm}$};
% Dimension: 20 cm on bottom (AF)
\draw[<->] (0,-3) -- (20,-3);
\node[below] at (10,-3) {$20\,\mathrm{cm}$};
\end{tikzpicture}
$A B C D E F$ is the L -shaped cross-section of a uniform solid. This cross-section passes through the centre of mass of the solid and has dimensions as shown in Fig. 1.\\
(i) Find the distance of the centre of mass of the solid from the edge $A B$ of the cross-section.
\begin{tikzpicture}[scale=0.2]
\coordinate (A) at (0,0);
\coordinate (B) at (0,40);
\coordinate (C) at (10,40);
\coordinate (D) at (10,10);
\coordinate (E) at (20,10);
\coordinate (F) at (20,0);
\coordinate (dir) at (4, 2.2);
\fill[gray!60] (-40, -15) -- (20, -15) -- (80, 25) -- (25, 25) -- cycle;
\filldraw[fill=white, thick] ($(A)+4*(dir)$) -- ($(B)+4*(dir)$) -- ($(C)+4*(dir)$) -- ($(D)+4*(dir)$) -- ($(E)+4*(dir)$) -- ($(F)+4*(dir)$) -- cycle;
\filldraw[fill=white, thick] ($(B)-4*(dir)$) -- ($(C)-4*(dir)$) -- ($(C)+4*(dir)$) -- ($(B)+4*(dir)$) -- cycle;
\filldraw[fill=white, thick] ($(C)-4*(dir)$) -- ($(D)-4*(dir)$) -- ($(D)+4*(dir)$) -- ($(C)+4*(dir)$) -- cycle;
\filldraw[fill=white, thick] ($(D)-4*(dir)$) -- ($(E)-4*(dir)$) -- ($(E)+4*(dir)$) -- ($(D)+4*(dir)$) -- cycle;
\filldraw[fill=white, thick] ($(E)-4*(dir)$) -- ($(F)-4*(dir)$) -- ($(F)+4*(dir)$) -- ($(E)+4*(dir)$) -- cycle;
\filldraw[fill=white, thick] ($(A)-4*(dir)$) -- ($(B)-4*(dir)$) -- ($(C)-4*(dir)$) -- ($(D)-4*(dir)$) -- ($(E)-4*(dir)$) -- ($(F)-4*(dir)$) -- cycle;
\draw[dashed, thick] (B) -- (C) -- (D) -- (E) -- (F);
\draw ($(B)-(10,0)$) -- (B);
\draw[thick, ->] ($(B)-(20,0)$) -- ($(B)-(10,0)$) node[above, midway] {$P$ N};
\node[above left] at (B) {$B$};
\node[above] at (C) {$C$};
\node[left] at (D) {$D$};
\node[above] at (E) {$E$};
\node[above right] at (F) {$F$};
\end{tikzpicture}
The solid rests in equilibrium with the face containing the edge $A F$ of the cross-section in contact with a horizontal table. The weight of the solid is $W$ N. A horizontal force of magnitude $P$ N is applied to the solid at the point $B$, in the direction of $B C$ (see Fig. 2). The table is sufficiently rough to prevent sliding.\\
(ii) Find $P$ in terms of $W$, given that the equilibrium of the solid is about to be broken.
\hfill \mbox{\textit{CAIE M2 2005 Q3 [6]}}