OCR Further Mechanics 2019 June — Question 7 12 marks

Exam BoardOCR
ModuleFurther Mechanics (Further Mechanics)
Year2019
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCircular Motion 2
TypeParticle on outer surface of sphere
DifficultyChallenging +1.2 This is a standard Further Maths mechanics problem involving a particle on a smooth hemisphere. Part (a) requires finding the point where the particle leaves the surface (using N=0 and energy conservation), then projectile motion to find landing distance - a well-established multi-step procedure. Part (b) is a straightforward conceptual check. While requiring several techniques, this follows a standard template that Further Maths students practice extensively.
Spec3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model6.02i Conservation of energy: mechanical energy principle
7 \includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-4_330_1061_989_267} The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere. A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\). \(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).
  1. Find an exact expression for the distance \(O F\). The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
  2. Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth. \section*{END OF QUESTION PAPER}
Question 7:
AnswerMarks Guidance
7(a) 1
mv2 mrcosmrcos oe
2
3
v2 2rcos r
2
mcos (C)ma
v2
a
r
a v2
cos 
 r
v2  1r
2
 1  1
rcos rsint t2
 
 2  2
3r r
t2  t 0
2 
1 r  
t 22 6
4 
1 1 r  
OF rsin rcos 22 6
2 4 
r 
OF  113 3 oe
AnswerMarks
8M1
A1
B1
B1
M1
A1
M1
*M1
Dep*
M1
Dep*
M1
A1
AnswerMarks
[11]3.3
1.1
1.1
1.1
2.2a
2.2a
3.4
1.1
1.1
3.4
AnswerMarks
1.1Conservation of energy
NII for P at point where it is about
to lose contact with surface.
soi
Use previous two results to relate v
and cos θ
1
for v or v2 or cos
2
Use of sut1at2using their
2
vertical component of v as u where
v has come from consideration of
theta
Reduction to 3 term quadratic with
numerical values for trig ratios
Solve for t (BC)
AnswerMarks
Find OFθ is the angle between OP
and the upward vertical
Could see contact force, C,
later set to 0
Or use of trajectory eqn:
x2
yxtan with
2v2cos2
yrcos and v2  1r
2
8x2 2 3rxr2 0
11 3
x r
8
11 3
rsin r
8
AnswerMarks Guidance
7(b) Unchanged since OF does not depend on 
[1]3.5a
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Question 7:
7 | (a) | 1
mv2 mrcosmrcos oe
2
3
v2 2rcos r
2
mcos (C)ma
v2
a
r
a v2
cos 
 r
v2  1r
2
 1  1
rcos rsint t2
 
 2  2
3r r
t2  t 0
2 
1 r  
t 22 6
4 
1 1 r  
OF rsin rcos 22 6
2 4 
r 
OF  113 3 oe
8 | M1
A1
B1
B1
M1
A1
M1
*M1
Dep*
M1
Dep*
M1
A1
[11] | 3.3
1.1
1.1
1.1
2.2a
2.2a
3.4
1.1
1.1
3.4
1.1 | Conservation of energy
NII for P at point where it is about
to lose contact with surface.
soi
Use previous two results to relate v
and cos θ
1
for v or v2 or cos
2
Use of sut1at2using their
2
vertical component of v as u where
v has come from consideration of
theta
Reduction to 3 term quadratic with
numerical values for trig ratios
Solve for t (BC)
Find OF | θ is the angle between OP
and the upward vertical
Could see contact force, C,
later set to 0
Or use of trajectory eqn:
x2
yxtan with
2v2cos2
yrcos and v2  1r
2
8x2 2 3rxr2 0
11 3
x r
8
11 3
rsin r
8
7 | (b) | Unchanged since OF does not depend on  | E1
[1] | 3.5a
PMT
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
© OCR 2019
7\\
\includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-4_330_1061_989_267}

The flat surface of a smooth solid hemisphere of radius $r$ is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by $\gamma$. $O$ is the centre of the flat surface of the hemisphere.

A particle $P$ is held at a point on the surface of the hemisphere such that the angle between $O P$ and the upward vertical through $O$ is $\alpha$, where $\cos \alpha = \frac { 3 } { 4 }$.\\
$P$ is then released from rest. $F$ is the point on the plane where $P$ first hits the plane (see diagram).
\begin{enumerate}[label=(\alph*)]
\item Find an exact expression for the distance $O F$.

The acceleration due to gravity on and near the surface of the planet Earth is roughly $6 \gamma$.
\item Explain whether $O F$ would increase, decrease or remain unchanged if the action were repeated on the planet Earth.

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

\hfill \mbox{\textit{OCR Further Mechanics 2019 Q7 [12]}}
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