| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Moments |
| Type | Beam suspended by vertical ropes |
| Difficulty | Standard +0.3 This is a standard mechanics statics problem requiring moment equilibrium about a point and resolving forces vertically. Part (a) involves setting up two equations (moments and forces) with the given tension relationship to find the center of mass—straightforward but requires careful bookkeeping across 5 marks. Part (b) is a simple application of moment = force × distance with the constraint that the particle must be on the beam. While multi-step, these are routine A-level mechanics techniques with no novel insight required, making it slightly easier than average. |
| Spec | 3.04b Equilibrium: zero resultant moment and force |
\begin{tikzpicture}[>=latex]
\pgfmathsetmacro{\s}{2}
\pgfmathsetmacro{\xA}{0}
\pgfmathsetmacro{\xC}{0.5*\s}
\pgfmathsetmacro{\xD}{3.3*\s}
\pgfmathsetmacro{\xB}{4*\s}
\pgfmathsetmacro{\rodH}{3}
% Beam
\fill[gray!40] (\xA,-0.1) rectangle (\xB,0.1);
\draw[thick] (\xA,-0.1) rectangle (\xB,0.1);
% Vertical rod at C
\draw[thick] (\xC,0.1) -- (\xC,\rodH);
\fill (\xC,\rodH) circle (4pt);
% Vertical rod at D
\draw[thick] (\xD,0.1) -- (\xD,\rodH);
\fill (\xD,\rodH) circle (4pt);
% Point labels
\node[below=6pt] at (\xA, -0.1) {$A$};
\node[below=6pt] at (\xC, -0.1) {$C$};
\node[below=6pt] at (\xD, -0.1) {$D$};
\node[below=6pt] at (\xB, -0.1) {$B$};
% 0.5 m label
\node[above=4pt] at ({(\xA+\xC)/2}, 0.1) {0.5\,m};
% 0.7 m label
\node[above=4pt] at ({(\xD+\xB)/2}, 0.1) {0.7\,m};
% 4 m dimension arrow
\draw[<->] (\xA,-0.8) -- node[below] {4\,m} (\xB,-0.8);
\end{tikzpicture}
A beam, $AB$, has length 4 m and mass 20 kg. The beam is suspended horizontally by two vertical ropes. One rope is attached to the beam at $C$, where $AC = 0.5$ m. The other rope is attached to the beam at $D$, where $DB = 0.7$ m (see diagram).
The beam is modelled as a non-uniform rod and the ropes as light inextensible strings.
It is given that the tension in the rope at $C$ is three times the tension in the rope at $D$.
\begin{enumerate}[label=(\alph*)]
\item Determine the distance of the centre of mass of the beam from $A$. [5]
\end{enumerate}
A particle of mass $m$ kg is now placed on the beam at a point where the magnitude of the moment of the particle's weight about $C$ is 3.5$mg$ N m. The beam remains horizontal and in equilibrium.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine the largest possible value of $m$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2021 Q12 [7]}}