Standard +0.3 This is a standard M1 statics problem involving resolving forces in equilibrium. Part (i) requires basic vertical resolution to establish an inequality. Part (ii) involves routine trigonometry to find angles, then resolving forces to find tension and weight. Part (iii) is straightforward reasoning about the limiting case. All techniques are standard textbook exercises with no novel insight required, making it slightly easier than average.
\includegraphics{figure_6}
A smooth ring \(R\) of weight \(W\) N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). A horizontal force of magnitude \(P\) N acts on \(R\). The system is in equilibrium with the string taut; \(AR\) makes an angle \(\alpha\) with the downward vertical and \(BR\) makes an angle \(\beta\) with the upward vertical (see Fig. 1).
By considering the vertical components of the forces acting on \(R\), show that \(\alpha < \beta\). [3]
\includegraphics{figure_6ii}
It is given that when \(P = 14\), \(AR = 0.4\) m, \(BR = 0.3\) m and the distance of \(R\) from the vertical line \(AB\) is \(0.24\) m (see Fig. 2). Find
Particles P and Q, of masses m kg and 0.05 kg respectively, are attached to the ends of a light inextensible
string which passes over a smooth pulley. Q is attached to a particle R of mass 0.45 kg by a light inextensible
string. The strings are taut, and the portions of the strings not in contact with the pulley are vertical. P is in
contact with a horizontal surface when the particles are released from rest (see diagram). The tension in the
string QR is 2.52 N during the descent of R.
(i) (a) Find the acceleration of R during its descent. [2]
(b) By considering the motion of Q, calculate the tension in the string PQ during the descent of R. [3]
(ii) Find the value of m. [3]
R strikes the surface 0.5 s after release and does not rebound. During their subsequent motion, P does not
reach the pulley and Q does not reach the surface.
(iii) Calculate the greatest height of P above the surface. [8]
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\includegraphics{figure_6}
A smooth ring $R$ of weight $W$ N is threaded on a light inextensible string. The ends of the string are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$. A horizontal force of magnitude $P$ N acts on $R$. The system is in equilibrium with the string taut; $AR$ makes an angle $\alpha$ with the downward vertical and $BR$ makes an angle $\beta$ with the upward vertical (see Fig. 1).
\begin{enumerate}[label=(\roman*)]
\item By considering the vertical components of the forces acting on $R$, show that $\alpha < \beta$. [3]
\end{enumerate}
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item \includegraphics{figure_6ii}
It is given that when $P = 14$, $AR = 0.4$ m, $BR = 0.3$ m and the distance of $R$ from the vertical line $AB$ is $0.24$ m (see Fig. 2). Find
\begin{enumerate}[label=(\alph*)]
\item the tension in the string, [3]
\item the value of $W$. [3]
\end{enumerate}
\item For the case when $P = 0$,
\begin{enumerate}[label=(\alph*)]
\item describe the position of $R$, [1]
\item state the tension in the string. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR M1 Q6 [11]}}