Standard +0.3 This is a standard M1 mechanics question involving Newton's second law applied to connected particles. Parts (i)-(iii) are straightforward applications of F=ma to the whole system and subsystems. Parts (iv)-(v) require careful consideration of forces when coupling Y becomes slack, but follow standard problem-solving techniques taught in M1. The multi-part structure and bookwork nature make it slightly easier than average.
\includegraphics{figure_6}
A train of total mass \(80000\) kg consists of an engine \(E\) and two trucks \(A\) and \(B\). The engine \(E\) and truck \(A\) are connected by a rigid coupling \(X\), and trucks \(A\) and \(B\) are connected by another rigid coupling \(Y\). The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on \(E\), \(A\) and \(B\) are \(10500\) N, \(3000\) N and \(1500\) N respectively (see diagram).
By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than \(15000\) N. [2]
Show that, when the magnitude of the driving force is \(35000\) N, the acceleration of the train is \(0.25\) m s\(^{-2}\). [2]
Hence find the mass of \(E\), given that the tension in the coupling \(X\) is \(8500\) N when the magnitude of the driving force is \(35000\) N. [3]
The driving force is replaced by a braking force of magnitude \(15000\) N acting on the engine. The force exerted by the coupling \(Y\) is zero.
Find the mass of \(B\). [5]
Show that the coupling \(X\) exerts a forward force of magnitude \(1500\) N on the engine. [2]
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 train of total mass $80000$ kg consists of an engine $E$ and two trucks $A$ and $B$. The engine $E$ and truck $A$ are connected by a rigid coupling $X$, and trucks $A$ and $B$ are connected by another rigid coupling $Y$. The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on $E$, $A$ and $B$ are $10500$ N, $3000$ N and $1500$ N respectively (see diagram).
\begin{enumerate}[label=(\roman*)]
\item By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than $15000$ N. [2]
\item Show that, when the magnitude of the driving force is $35000$ N, the acceleration of the train is $0.25$ m s$^{-2}$. [2]
\item Hence find the mass of $E$, given that the tension in the coupling $X$ is $8500$ N when the magnitude of the driving force is $35000$ N. [3]
\end{enumerate}
The driving force is replaced by a braking force of magnitude $15000$ N acting on the engine. The force exerted by the coupling $Y$ is zero.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Find the mass of $B$. [5]
\item Show that the coupling $X$ exerts a forward force of magnitude $1500$ N on the engine. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR M1 Q6 [14]}}