OCR MEI Further Mechanics A AS 2019 June — Question 6 11 marks

Exam BoardOCR MEI
ModuleFurther Mechanics A AS (Further Mechanics A AS)
Year2019
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMomentum and Collisions 1
TypeThree-particle sequential collisions
DifficultyStandard +0.3 This is a standard sequential collision problem requiring systematic application of conservation of momentum and restitution equations twice, followed by a routine energy loss calculation. While it involves multiple steps and careful bookkeeping across three particles, the techniques are entirely standard for Further Mechanics with no novel insights required. The second part about the rod is a typical statics problem with standard resolution of forces and moments.
Spec3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts
6 Three particles, A, B and C are in a straight line on a smooth horizontal surface.
The particles have masses \(5 \mathrm {~kg} , 3 \mathrm {~kg}\) and 1 kg respectively. Particles B and C are at rest. Particle A is projected towards B with a speed of \(u \mathrm {~ms} ^ { - 1 }\) and collides with B . The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Particle B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 3 }\).
  1. Determine whether any further collisions occur.
  2. Given that the loss of kinetic energy during the initial collision between A and B is 4.8 J , find the value of \(u\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-5_607_501_294_301} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} Fig. 7 shows a uniform rod AB of length \(4 a\) and mass \(m\).
    The end A rests against a rough vertical wall. A light inextensible string is attached to the rod at B and to a point C on the wall vertically above A , where \(\mathrm { AC } = 4 a\). The plane ABC is perpendicular to the wall and the angle ABC is \(30 ^ { \circ }\). The system is in limiting equilibrium. Find the coefficient of friction between the wall and the rod. \section*{END OF QUESTION PAPER}
Question 6:
AnswerMarks Guidance
6(a) Let the speeds of A and B after the collision
be u and u
AnswerMarks
A BAllow sign errors for all
M marks
5u5u 3u
AnswerMarks Guidance
A BM1 3.3
– correct number of termsAllow wrong mass for
linear momentum marks
u u 1u
AnswerMarks Guidance
A B 3M1 3.3
terms and consistent with CLMNo masses for NEL
marks
u  1u, u  5u
AnswerMarks Guidance
A 2 B 6A1 1.1
correctlyNote: can be done in
terms of, for example, u B
w & w − speeds of B & C after
B C
 
3 5u 3w w
AnswerMarks Guidance
6 B CM1 1.1
- correct number of termsCan use u or their u
B B
 
w w 1 5u
AnswerMarks Guidance
B C 3 6M1 1.1
terms and consistent with CLMCan use u or their u
B B
w  5u, w  5u
AnswerMarks Guidance
B 9 C 6A1 1.1
No further collisions asw u anw w
AnswerMarks Guidance
B A C BA1 2.4
justificationAll vels must be correct to
gain this mark
Alternative Method: First A1 may be gained later by proving that 𝑤 > 𝑢
𝐵 𝐴
Second A1 may be gained by proving that 𝑤 > 𝑤
𝐶 𝐵
Third A1 is then gained for complete correct solution
[7]
AnswerMarks Guidance
(b)Initial KE = 5u2
2B1 1.1
Change = 5u2−(1 5 u 2 1 3 u 2)
AnswerMarks Guidance
2 2 A 2 BB1ft 1.1
1 1 1 2 1 5 2
.5𝑢2−( .5( 𝑢) + .3( 𝑢) )=4.8
AnswerMarks Guidance
2 2 2 2 6M1 1.1
correct number of termsMight be in terms of u B
Allow sign errors
AnswerMarks Guidance
u2.4A1 1.1
[4]
Question 6:
6 | (a) | Let the speeds of A and B after the collision
be u and u
A B | Allow sign errors for all
M marks
5u5u 3u
A B | M1 | 3.3 | Conservation of linear momentum
– correct number of terms | Allow wrong mass for
linear momentum marks
u u 1u
A B 3 | M1 | 3.3 | Use of NEL, correct number of
terms and consistent with CLM | No masses for NEL
marks
u  1u, u  5u
A 2 B 6 | A1 | 1.1 | Solve simultaneous equation
correctly | Note: can be done in
terms of, for example, u B
w & w − speeds of B & C after
B C
 
3 5u 3w w
6 B C | M1 | 1.1 | Conservation of linear momentum
- correct number of terms | Can use u or their u
B B
 
w w 1 5u
B C 3 6 | M1 | 1.1 | Use of NEL, correct number of
terms and consistent with CLM | Can use u or their u
B B
w  5u, w  5u
B 9 C 6 | A1 | 1.1
No further collisions asw u anw w
B A C B | A1 | 2.4 | Consider all 3 particles – values +
justification | All vels must be correct to
gain this mark
Alternative Method: First A1 may be gained later by proving that 𝑤 > 𝑢
𝐵 𝐴
Second A1 may be gained by proving that 𝑤 > 𝑤
𝐶 𝐵
Third A1 is then gained for complete correct solution
[7]
(b) | Initial KE = 5u2
2 | B1 | 1.1
Change = 5u2−(1 5 u 2 1 3 u 2)
2 2 A 2 B | B1ft | 1.1
1 1 1 2 1 5 2
.5𝑢2−( .5( 𝑢) + .3( 𝑢) )=4.8
2 2 2 2 6 | M1 | 1.1 | Setting up an equation in u with
correct number of terms | Might be in terms of u B
Allow sign errors
u2.4 | A1 | 1.1
[4]
6 Three particles, A, B and C are in a straight line on a smooth horizontal surface.\\
The particles have masses $5 \mathrm {~kg} , 3 \mathrm {~kg}$ and 1 kg respectively. Particles B and C are at rest. Particle A is projected towards B with a speed of $u \mathrm {~ms} ^ { - 1 }$ and collides with B . The coefficient of restitution between A and B is $\frac { 1 } { 3 }$.

Particle B subsequently collides with C. The coefficient of restitution between B and C is $\frac { 1 } { 3 }$.
\begin{enumerate}[label=(\alph*)]
\item Determine whether any further collisions occur.
\item Given that the loss of kinetic energy during the initial collision between A and B is 4.8 J , find the value of $u$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-5_607_501_294_301}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}

Fig. 7 shows a uniform rod AB of length $4 a$ and mass $m$.\\
The end A rests against a rough vertical wall. A light inextensible string is attached to the rod at B and to a point C on the wall vertically above A , where $\mathrm { AC } = 4 a$. The plane ABC is perpendicular to the wall and the angle ABC is $30 ^ { \circ }$.

The system is in limiting equilibrium.

Find the coefficient of friction between the wall and the rod.

\section*{END OF QUESTION PAPER}
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2019 Q6 [11]}}
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