OCR MEI Further Mechanics B AS 2019 June — Question 6 14 marks

Exam BoardOCR MEI
ModuleFurther Mechanics B AS (Further Mechanics B AS)
Year2019
SessionJune
Marks14
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCircular Motion 2
TypeParticle on outer surface of sphere
DifficultyChallenging +1.2 This is a standard Further Mechanics circular motion problem requiring energy conservation, circular motion equations (vΒ²/r = g cos ΞΈ + N/m), and projectile motion. Part (a) is a routine 'show that' using energy and the condition Nβ‰₯0 at H. Parts (b-d) follow standard procedures: finding loss of contact condition, then projectile motion. While multi-step with several marks, it follows well-established templates for this topic without requiring novel insight.
Spec3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model6.02d Mechanical energy: KE and PE concepts6.02e Calculate KE and PE: using formulae6.02i Conservation of energy: mechanical energy principle6.05a Angular velocity: definitions6.05d Variable speed circles: energy methods
6 A smooth solid hemisphere of radius \(a\) is fixed with its plane face in contact with a horizontal surface.
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass \(m\) is held at a point S on the surface of the hemisphere such that angle HOS is \(30 ^ { \circ }\), as shown in Fig. 6. The particle is projected from S with speed \(0.8 \sqrt { a g }\) along the surface of the hemisphere towards H . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-5_358_1056_497_244} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Show that the particle passes through H without leaving the surface of the hemisphere. After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle \(\mathrm { HOQ } = \theta ^ { \circ }\).
  2. State, in terms of \(g\) and \(\theta\), the tangential component of the acceleration of the particle when it is at Q . The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
  3. Find the value of \(\cos \theta\) correct to 3 significant figures.
  4. Show that \(\mathrm { OL } = k a\), where \(k\) is to be found correct to 3 significant figures.
Question 6:
AnswerMarks Guidance
6(a) π‘šΓ—0.64π‘Žπ‘”
At S, force towards centre = or
π‘Ž
AnswerMarks Guidance
Weight component along SO = π‘šπ‘”cos30Β°B1 2.1
energy first.
0.866π‘šπ‘” > 0.64π‘šπ‘” so P does not leave
AnswerMarks Guidance
hemisphereE1 1.1
Or
ο€½0.226mg ο€Ύ0
1
Energy at S = π‘šΓ—0.64π‘Žπ‘”+π‘šπ‘”π‘Žcos30Β°
AnswerMarks Guidance
2M1 2.1
This is greater than π‘šπ‘”π‘Ž (PE at top) so P
AnswerMarks Guidance
passes through HA1 1.1
[4]
AnswerMarks Guidance
(b)𝑔sinπœƒ B1
[1]
AnswerMarks
(c)π‘šπ‘£2
At Q, π‘šπ‘”cosπœƒ =
AnswerMarks Guidance
π‘ŽM1 3.3
1
π‘šΓ—0.64π‘Žπ‘”+π‘šπ‘”π‘Žcos30Β°
2
1
= π‘šπ‘£2 +π‘šπ‘”π‘Žcosπœƒ
AnswerMarks Guidance
2M1 1.1
Eliminate v (or cosπœƒ)M1 3.3
balance energy.
1
cosπœƒ = (0.64+2cos30Β°)
3
1
Or (0.64 3)
AnswerMarks Guidance
3A1 1.1
[4]
AnswerMarks Guidance
(d)𝑣2 = 7.748664π‘Ž M1
(c) (Method mark)
1
π‘Žcosπœƒ = 2.7836βˆšπ‘Žsinπœƒπ‘‘ + 𝑔𝑑2
AnswerMarks Guidance
2M1 3.4
height of π‘Žsinπœƒ
AnswerMarks Guidance
𝑑 = 0.2638βˆšπ‘ŽA1 1.1
OL = π‘Žsinπœƒ+2.7836βˆšπ‘Žcosπœƒ Γ—
AnswerMarks Guidance
(0.2638βˆšπ‘Ž)M1 3.4
1.19aA1 1.1
then allow answers in the
range [1.192,1.194]
[5]
PPMMTT
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Question 6:
6 | (a) | π‘šΓ—0.64π‘Žπ‘”
At S, force towards centre = or
π‘Ž
Weight component along SO = π‘šπ‘”cos30Β° | B1 | 2.1 | Candidates might consider
energy first.
0.866π‘šπ‘” > 0.64π‘šπ‘” so P does not leave
hemisphere | E1 | 1.1 | Rο€½mgcos30ο‚°ο€­0.64mg
Or
ο€½0.226mg ο€Ύ0
1
Energy at S = π‘šΓ—0.64π‘Žπ‘”+π‘šπ‘”π‘Žcos30Β°
2 | M1 | 2.1
This is greater than π‘šπ‘”π‘Ž (PE at top) so P
passes through H | A1 | 1.1 | Or show that v2 is positive at H | v2 = 0.372ag
[4]
(b) | 𝑔sinπœƒ | B1 | 1.2
[1]
(c) | π‘šπ‘£2
At Q, π‘šπ‘”cosπœƒ =
π‘Ž | M1 | 3.3
1
π‘šΓ—0.64π‘Žπ‘”+π‘šπ‘”π‘Žcos30Β°
2
1
= π‘šπ‘£2 +π‘šπ‘”π‘Žcosπœƒ
2 | M1 | 1.1
Eliminate v (or cosπœƒ) | M1 | 3.3 | Must be in an attempt to
balance energy.
1
cosπœƒ = (0.64+2cos30Β°)
3
1
Or (0.64 3)
3 | A1 | 1.1 | 0.79068…; or ΞΈ = 37.75Β°
[4]
(d) | 𝑣2 = 7.748664π‘Ž | M1 | 1.1 | Or 0.791ag; or v = 2.7836√a | Follow through from part
(c) (Method mark)
1
π‘Žcosπœƒ = 2.7836βˆšπ‘Žsinπœƒπ‘‘ + 𝑔𝑑2
2 | M1 | 3.4 | Motion vertically | BOD for initial vertical
height of π‘Žsinπœƒ
𝑑 = 0.2638βˆšπ‘Ž | A1 | 1.1 | Condone sight of βˆ’0.611
OL = π‘Žsinπœƒ+2.7836βˆšπ‘Žcosπœƒ Γ—
(0.2638βˆšπ‘Ž) | M1 | 3.4
1.19a | A1 | 1.1 | 1.1922a | If more than 3 s.f shown
then allow answers in the
range [1.192,1.194]
[5]
PPMMTT
OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
Cambridge
CB2 8EA
OCR Customer Contact Centre
Education and Learning
Telephone: 01223 553998
Facsimile: 01223 552627
Email: general.qualifications@ocr.org.uk
www.ocr.org.uk
For staff training purposes and as part of our quality assurance
programme your call may be recorded or monitored
Oxford Cambridge and RSA Examinations
is a Company Limited by Guarantee
Registered in England
Registered Office; The Triangle Building, Shaftesbury Road, Cambridge, CB2 8EA
Registered Company Number: 3484466
OCR is an exempt Charity
OCR (Oxford Cambridge and RSA Examinations)
Head office
Telephone: 01223 552552
Facsimile: 01223 552553
Β© OCR 2019
6 A smooth solid hemisphere of radius $a$ is fixed with its plane face in contact with a horizontal surface.\\
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass $m$ is held at a point S on the surface of the hemisphere such that angle HOS is $30 ^ { \circ }$, as shown in Fig. 6. The particle is projected from S with speed $0.8 \sqrt { a g }$ along the surface of the hemisphere towards H .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-5_358_1056_497_244}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Show that the particle passes through H without leaving the surface of the hemisphere.

After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle $\mathrm { HOQ } = \theta ^ { \circ }$.
\item State, in terms of $g$ and $\theta$, the tangential component of the acceleration of the particle when it is at Q .

The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
\item Find the value of $\cos \theta$ correct to 3 significant figures.
\item Show that $\mathrm { OL } = k a$, where $k$ is to be found correct to 3 significant figures.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics B AS 2019 Q6 [14]}}
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