OCR MEI Further Mechanics A AS 2019 June — Question 2 12 marks

Exam BoardOCR MEI
ModuleFurther Mechanics A AS (Further Mechanics A AS)
Year2019
SessionJune
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDimensional Analysis
TypeDerive dimensions from formula
DifficultyModerate -0.3 This is a straightforward dimensional analysis question requiring standard techniques: deriving dimensions from a formula, checking dimensional consistency, a numerical calculation, and finding exponents. All parts follow routine procedures taught in Further Mechanics with no novel problem-solving required, making it slightly easier than average.
Spec6.01a Dimensions: M, L, T notation6.01b Units vs dimensions: relationship6.01c Dimensional analysis: error checking6.01d Unknown indices: using dimensions6.01e Formulate models: dimensional arguments
2 The surface tension of a liquid allows a metal needle to be at rest on the surface of the liquid.
The greatest mass \(m\) of a needle of length \(l\) which can be supported in this way by a liquid of surface tension \(S\) is given by the formula \(m = \frac { 2 S l } { g }\) where \(g\) is the acceleration due to gravity.
  1. Determine the dimensions of surface tension. Surface tension also allows liquids to rise up capillary tubes. Molly is experimenting with liquids in capillary tubes and she arrives at the formula \(h = \frac { 2 S } { \rho g r }\), where \(h\) is the height to which a liquid of surface tension \(S\) rises, \(\rho\) is the density of the liquid, and \(r\) is the radius of the capillary tube.
  2. Show that the equation for \(h\) is dimensionally consistent. In SI units, the surface tension of mercury is \(0.475 \mathrm {~kg} \mathrm {~s} ^ { - 2 }\) and its density is \(13500 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).
  3. Find the diameter of a capillary tube in which mercury will rise to a height of 10 cm . In another experiment, Molly finds that when liquid of surface tension \(S\) is poured onto a horizontal surface, puddles of depth \(d\) are formed. For this experiment she finds that \(d = k S ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }\) where \(k\) is a dimensionless constant.
  4. Determine the values of \(\alpha , \beta\) and \(\gamma\).
Question 2:
AnswerMarks Guidance
2(a) gLT2
mg M  LT2 
S 
 
 2l  L
AnswerMarks
SMT2B1
M1
AnswerMarks
A11.2
3.4
AnswerMarks
2.2aCorrect dimensions of g soi
Correctly substitution for m and l.
Allow g or ‘their’ g.
AnswerMarks
www(2)[S]L
Accept M =
LT−2
Sc B1 if done using units
[3]
AnswerMarks
(b)[𝜌] = ML−3
MT−2
RHS = = L
(ML−3)(LT−2)(L)
AnswerMarks
State same as LHS = LB1
B1
AnswerMarks
B11.2
1.1
AnswerMarks
1.1Dimensions of density soi
RHS correctly shown
Sc Allow if consistent errors in g
and S which cancel out
www
Sc allow full marks for proof that
AnswerMarks
units are consistentFt from (a)
Dependent on first two
marks
[3]
AnswerMarks
(c)2S
r 
gh
20.475
13 5009.80.1
7.18105
Depth = 1.4(4)×10−4 (m)
19
Or m
AnswerMarks
132300M1
A13.4
1.1Correct substitution into correct
equation for r
cao (oe e.g. 0.0143613…(cm))
SC If 0, allow B1 for figs (72) or
AnswerMarks
figs (14(4)) as final answer.Accept cm or m answer
without units; accept mm
if specified
[2]
AnswerMarks
(d)L =  MT2  ML3  LT2 
0, 31, 220
1 1 1
 ,  , 
AnswerMarks
2 2 2M1
M1
A1
AnswerMarks
A13.3
1.1
1.1
AnswerMarks
2.2aSet up with correct substitutions
Compare coeffs of M, L and T
All 3 equations correct
AnswerMarks
All correct wwwAllow M marks with their
S and g
All three needed
[4]
Question 2:
2 | (a) | gLT2
mg M  LT2 
S 
 
 2l  L
SMT2 | B1
M1
A1 | 1.2
3.4
2.2a | Correct dimensions of g soi
Correctly substitution for m and l.
Allow g or ‘their’ g.
www | (2)[S]L
Accept M =
LT−2
Sc B1 if done using units
[3]
(b) | [𝜌] = ML−3
MT−2
RHS = = L
(ML−3)(LT−2)(L)
State same as LHS = L | B1
B1
B1 | 1.2
1.1
1.1 | Dimensions of density soi
RHS correctly shown
Sc Allow if consistent errors in g
and S which cancel out
www
Sc allow full marks for proof that
units are consistent | Ft from (a)
Dependent on first two
marks
[3]
(c) | 2S
r 
gh
20.475

13 5009.80.1
7.18105
Depth = 1.4(4)×10−4 (m)
19
Or m
132300 | M1
A1 | 3.4
1.1 | Correct substitution into correct
equation for r
cao (oe e.g. 0.0143613…(cm))
SC If 0, allow B1 for figs (72) or
figs (14(4)) as final answer. | Accept cm or m answer
without units; accept mm
if specified
[2]
(d) | L =  MT2  ML3  LT2 
0, 31, 220
1 1 1
 ,  , 
2 2 2 | M1
M1
A1
A1 | 3.3
1.1
1.1
2.2a | Set up with correct substitutions
Compare coeffs of M, L and T
All 3 equations correct
All correct www | Allow M marks with their
S and g
All three needed
[4]
2 The surface tension of a liquid allows a metal needle to be at rest on the surface of the liquid.\\
The greatest mass $m$ of a needle of length $l$ which can be supported in this way by a liquid of surface tension $S$ is given by the formula\\
$m = \frac { 2 S l } { g }$\\
where $g$ is the acceleration due to gravity.
\begin{enumerate}[label=(\alph*)]
\item Determine the dimensions of surface tension.

Surface tension also allows liquids to rise up capillary tubes. Molly is experimenting with liquids in capillary tubes and she arrives at the formula $h = \frac { 2 S } { \rho g r }$, where $h$ is the height to which a liquid of surface tension $S$ rises, $\rho$ is the density of the liquid, and $r$ is the radius of the capillary tube.
\item Show that the equation for $h$ is dimensionally consistent.

In SI units, the surface tension of mercury is $0.475 \mathrm {~kg} \mathrm {~s} ^ { - 2 }$ and its density is $13500 \mathrm {~kg} \mathrm {~m} ^ { - 3 }$.
\item Find the diameter of a capillary tube in which mercury will rise to a height of 10 cm .

In another experiment, Molly finds that when liquid of surface tension $S$ is poured onto a horizontal surface, puddles of depth $d$ are formed. For this experiment she finds that\\
$d = k S ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }$\\
where $k$ is a dimensionless constant.
\item Determine the values of $\alpha , \beta$ and $\gamma$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2019 Q2 [12]}}
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