Moments and centre of mass
P3
64 minutes
64 marks
Page 1 of 28
Q1.
(a)
The diagram shows a pendulum.
Draw an X on the diagram above, so that the centre of the X marks the centre of mass of
the pendulum bob.
(1)
(b)
A large clock keeps time using the swing of a pendulum.
(i)
The frequency of the swinging pendulum is 0.5 hertz.
Calculate the periodic time of the pendulum.
Use the correct equation from the Physics Equations Sheet.
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Periodic time = .............................. seconds
(2)
Page 2 of 28
(ii)
Calculate the number of complete swings the pendulum would make in 60 seconds.
Use your answer from part (b)(i) in your calculation.
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Number of swings in 60 seconds = ..............................
(2)
(c)
The diagram shows a clock on a trolley.
The trolley is being used to move the clock.
Calculate the moment of the 64 N force about the pivot.
Use the correct equation from the Physics Equations Sheet.
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Moment of the force = .............................. Nm
(2)
Page 3 of 28
(d)
The design of the trolley is now changed to make it taller.
How does making the trolley taller affect the moment produced by the 64 N force about the
pivot?
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(1)
(Total 8 marks)
Page 4 of 28
Q2.
A company makes a wheel wrench with an extending handle. The company claims that the
extending handle makes it easier to loosen the wheel nuts on a car.
The diagram shows the wheel wrench being used without the handle extended.
(a)
(i)
Use the equation in the box to calculate the moment produced by the force on the
wrench.
moment
=
force
× perpendicular distance from the line of
action of the force to the axis of rotation
Show clearly how you work out your answer.
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Moment = ............................................................. newton metres
(2)
(ii)
Units can be written in words or symbols.
Which of the following is the unit for a moment written using symbols?
Draw a ring around your answer.
nm
Nm
nM
NM
(1)
Page 5 of 28
(b)
The wheel nut will not move and so the handle of the wrench is extended.
It is now easy to loosen the wheel nut using the same force as before.
Explain why.
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(2)
(Total 5 marks)
Q3.
(a)
Every object has a centre of mass. What is meant by the centre of mass?
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(1)
(b)
The drawing shows a thin sheet of plastic. The sheet is 250 mm wide. Two holes, each
with a radius of 2 mm, have been drilled through the sheet.
Describe how you could use:
•
•
•
•
•
a clamp and stand
a steel rod 100 mm long and with a radius of I mm
a weight on a thin piece of string (= a plumb line)
a ruler
a pen which will write on the plastic sheet
to find the centre of mass of the plastic sheet.
Page 6 of 28
To gain full marks in this question you should write your ideas in good English. Put them
into a sensible order and use the correct scientific words.
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(5)
(c)
There is a trapdoor in the ceiling of a house.
The trapdoor weighs 44 N.
The drawing shows a side view of the trapdoor.
(i)
Complete the three spaces to give the equation which is used to calculate the
turning effect of a force.
......................... = ......................... × perpendicular between .........................
line of action and pivot
(1)
(ii)
Calculate the turning effect, about the hinge, due to the weight of the trapdoor.
Show clearly how you work out your final answer and give the unit.
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Turning effect = ..............................................
(3)
(Total 10 marks)
Page 7 of 28
Q4.
A student was asked to find the centre of mass of a thin sheet of card. The diagram shows
the result of the student’s experiment. The student drew two lines onto the card. The centre of
mass is where the two lines cross.
(a)
Describe how the student found the correct positions to draw the two lines.
You may include a labelled diagram in your answer.
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(3)
Page 8 of 28
(b)
Explain how the student can check that the position found for the centre of mass is
accurate.
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(2)
(Total 5 marks)
Q5.
Tractors are often used on sloping fields, so stability is important in their design.
On the diagram, the centre of the X marks the centre of mass of the tractor.
(a)
Explain why the tractor has not toppled over. You may add to the diagram to help you to
explain.
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(3)
Page 9 of 28
(b)
Give two features of the tractor which affect its stability and state how each feature could
be changed to increase the tractor’s stability.
Feature 1 .....................................................................................................................
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Feature 2 .....................................................................................................................
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(2)
(Total 5 marks)
Q6.
The diagrams show two concrete mixers.
Concrete mixer A
Concrete mixer B
On each diagram, the centre of the white X marks the centre of mass of the concrete mixer and
its contents.
(a)
Complete the sentence to explain what the term centre of mass means.
The centre of mass of a concrete mixer and its contents is ........................
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(1)
Page 10 of 28
(b)
Both diagrams are drawn to the same scale.
Concrete mixer B is more stable than concrete mixer A.
The two features which make concrete mixer B more stable are:
1 ......................................................................................................................
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2 ......................................................................................................................
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(2)
(c)
Use the terms ‘line of action of the weight’ and ‘resultant moment’ to explain why a stable
concrete mixer does not fall over when it is given a small push.
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(2)
(Total 5 marks)
Q7.
A student carries out an investigation using a metre rule as a pendulum.
(a)
Diagram 1 shows a metre rule.
Diagram 1
(i)
Draw, on Diagram 1, an X to show the position of the centre of mass of the rule.
(1)
(ii)
State what is meant by the ‘centre of mass of an object’.
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(1)
Page 11 of 28
(b)
The student taped a 100 g mass to a metre rule.
She set up the apparatus as shown in Diagram 2.
She suspended the metre rule from a nail through a hole close to one end, so she could
use the metre rule as a pendulum.
The distance d is the distance between the nail and the 100 g mass.
Diagram 2
(i)
Draw, on Diagram 2, a Y to show a possible position of the centre of mass of the
pendulum.
(1)
Page 12 of 28
(ii)
The student carried out an investigation to find out how the time period of the
pendulum varies with d.
Some of her results are shown in the table.
Time for 10 swings in seconds
d in cm
First
test
Second
test
Third
test
Mean
value
Mean time for
1 swing in
seconds
10.0
15.3
15.4
15.5
15.4
1.54
30.0
14.7
14.6
14.7
14.7
1.47
50.0
15.3
15.6
15.4
15.4
1.54
70.0
16.5
16.6
16.5
Complete the table.
You may use the space below to show your working.
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(3)
Page 13 of 28
(iii)
In this question you will be assessed on using good English, organising information
clearly and using specialist terms where appropriate.
Describe how the student would carry out the investigation to get the results in the
table in part (ii).
You should include:
•
any other apparatus required
•
how she should use the apparatus
•
how she could make it a fair test
•
a risk assessment
•
how she could make her results as accurate as possible.
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(6)
Page 14 of 28
(c)
A graph of the student’s results is shown below.
Distance d in cm
(i)
Describe the pattern shown by the graph.
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(2)
(ii)
The student thinks that the measurements of time for d = 10 cm might be
anomalous, so she takes a fourth measurement.
Her four measurements are shown below.
15.3 s
15.4 s
15.5 s
15.3 s
State whether you consider any of these measurements to be anomalous.
Justify your answer.
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(2)
(Total 16 marks)
Page 15 of 28
Q8.
Forces have different effects.
(a)
(i)
Use the correct answer from the box to complete the sentence.
slowing
stretching
turning
The moment of a force is the ............................................................... effect of the
force.
(1)
(ii)
What is meant by the centre of mass of an object?
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(1)
(b)
Some children build a see-saw using a plank of wood and a pivot.
The centre of mass of the plank is above the pivot.
Figure 1 shows a boy sitting on the see-saw. His weight is 400 N.
Figure 1
Calculate the anticlockwise moment of the boy in Nm.
Use the correct equation from Section A of the Physics Equations Sheet.
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Anticlockwise moment = ........................................ Nm
(2)
Page 16 of 28
(c)
Figure 2 shows a girl sitting at the opposite end of the see-saw. Her weight is 300 N.
Figure 2
The see-saw is now balanced.
The children move the plank. Its centre of mass, M, is now 0.25 m from the pivot as
shown in Figure 3.
Figure 3
The boy and girl sit on the see-saw as shown in Figure 3.
(i)
Describe and explain the rotation of the see-saw.
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(3)
Page 17 of 28
(ii)
The boy gets off the see-saw and a bigger boy gets on it in the same place. The girl
stays in the position shown in Figure 3. The plank is balanced. The weight of the
plank is 270 N.
Calculate the weight of the bigger boy.
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Weight of the bigger boy = ........................................ N
(3)
(Total 10 marks)
Page 18 of 28
M1.
(a)
centre of X drawn at centre of pendulum bob
judged by eye
accept dot drawn at centre of circle
1
(b)
(i)
2
allow 1 mark for correct substitution, ie
provided no
subsequent step shown
2
(ii)
30
or
60 ÷ their (b)(i) correctly calculated
allow 1 mark for
or
or 0.5 × 60
provided no subsequent step shown
2
(c)
51.2
allow 1 mark for correct substitution, ie 64 × 0.8 provided no
subsequent step shown
2
(d)
it increases (the moment)
must be comparative
accept 1 mark for calculation of the moment = 64 (Nm)
1
[8]
M2.
(a)
(i)
75
allow 1 mark for correct substitution ie 250 × 0.3
do not credit if subsequent step shown
allow 1 mark for an answer 7500
2
(ii)
Nm
1
(b)
force is (applied) further from the nut / pivot / axis of rotation
handle is longer is insufficient
do not accept less force needed
1
moment (on wrench) is larger
1
[5]
Page 19 of 28
M3.
(a)
point at which its mass (seems to) act or point at which gravity (seems to) act
accept ... its weight acts
accept correct statements if the intent is clear e.g.. .. if suspended,
the centre of gravity will be directly under the point of suspension
e.g.... (if the object is symmetrical), the centre of gravity is on the
or an axis (of symmetry)
do not credit just 'it is a point'
1
(b)
The answer to this question requires good English in a sensible order with correct use of
scientific terms. Quality of written communication should be considered in crediting points
in the mark scheme
maximum of 4 marks if ideas not well expressed
any five from:
clamp (steel) rod (horizontally)
no marks if method quite unworkable
hang plastic / sheet by rod through (one) hole
hang plumb line from rod
mark ends of plumb line on the sheet and
use the ruler to draw a straight line
repeat with other hole
centre of mass is where the lines cross
check by balancing at this point
maximum of 3 marks if no 'repeat with other hole'
5
(c)
(i)
(turning) effect or moment
force
distance
all three correct
accept weight
accept length
1
(ii)
17.6
allow 44 x 0.4 or 0.4 x 44 for 1 mark
2
Nm or newton metre(s)
do not accept N/m or N/cm
1760 Ncm gains all 3 marks
1
[10]
Page 20 of 28
M4.
Resource currently unavailable
M5.
(a)
(line of action of) its weight
1
falls inside its wheel base
accept ‘falls between the wheels’
the first two points may be credited by adding a vertical line from
the centre of the X on the diagram (1)
and labelling it weight / force / with a downwards arrow (1)
provided there is no contradiction between what is added to the
diagram and anything which may be written
1
(so there is) no (resultant / clockwise) moment / turning effect
1
(b)
centre of mass should be lower
accept ‘… centre of gravity’
accept ‘weight / mass low down’
not just ‘lower the roof’
1
wheel base should be wider
accept ‘long axle(s)’ for ‘wide wheel base’
allow bigger / larger wheel base
do not credit ‘long wheel base’
responses in either order
1
[5]
M6.
(a)
the point at which the (total) mass seems to act / appears to be concentrated
accept ‘weight’ for ‘mass’
accept the point at which gravity seems to act
do not accept a definitive statement eg where (all) the mass is
1
(b)
wider / larger base
marks are for a correct comparison
1
lower centre of mass
accept lower centre of gravity / c of g
1
(c)
line of action (of the weight) lies / falls inside the base
in each case the underlined term must be used correctly to gain
the mark
1
Page 21 of 28
the resultant moment returns mixer to its original position
accept there is no resultant moment / resultant moment is zero
accept resulting moment for resultant moment
do not accept converse argument
1
[5]
M7.
(a)
(i)
X placed at 50 cm mark
1
(ii)
point at which mass of object may be (thought to be) concentrated
1
(b)
(i)
Y placed between the centre of the rule and the upper part of mass
1
(ii)
16.5
allow for 1 mark
(16.5 + 16.6 +16.5) / 3
2
1.65
value consistent with mean value given
only penalise significant figures once
1
Page 22 of 28
(iii)
Marks awarded for this answer will be determined by the quality of
communication as well as the standard of the scientific response. Examiners
should apply a ‘best-fit’ approach to the marking.
0 marks
No relevant content
Level 1 (1 – 2 marks)
A description of a method which would provide results which may not be valid
Level 2 (3 – 4 marks)
A clear description of a method enabling some valid results to be obtained. A
safety factor is mentioned
Level 3 (5 – 6 marks)
A clear and detailed description of experiment. A safety factor is mentioned.
Uncertainty is mentioned
examples of the physics points made in the response:
additional apparatus
•
stopwatch
use of apparatus
•
•
•
•
•
measure from hole to centre of the mass
pull rule to one side, release
time for 10 swings and repeat
divide mean by 10
change position of mass and repeat
fair test
•
•
keep other factors constant
time to same point on swing
risk assessment
•
•
•
injury from sharp nail
stand topple over
rule hit someone
accuracy
•
•
•
•
•
take more than 4 values of d
estimate position of centre of slotted mass
small amplitudes
discard anomalous results
use of fiducial marker
6
(c)
(i)
initial reduction in T (reaching minimum value) as d increases
1
after 30 cm T increases for higher value of d
1
Page 23 of 28
(ii)
(no)
any two from:
•
•
•
fourth reading is close to mean
range of data 0.2 s / very small
variation in data is expected
2
[16]
M8.
(a)
(i)
turning
accept turning ringed in the box
1
(ii)
point at which mass (or weight) may be thought to be concentrated
accept the point from which the weight appears to act
allow focused for concentrated
do not accept most / some of the mass
do not accept region / area for point
1
(b)
600 (Nm)
400 × 1.5 gains 1 mark provided no subsequent steps shown
2
(c)
(i)
plank rotates clockwise
accept girl moves downwards
do not accept rotates to the right
1
(total) CM > (total) ACM
accept moment is larger on the girl’s side
1
weight of see-saw provides CM
answer must be in terms of moment
maximum of 2 marks if there is no reference to the weight of the
see-saw
1
(ii)
W = 445 (N)
W × 1.5 = (270 × 0.25) + (300 × 2.0) gains 2 marks
allow for 1 mark:
total CM = total ACM either stated or implied
or
(270 × 0.25) + (300 × 2.0)
if no other marks given
3
[10]
Page 24 of 28
E1.
(a) The majority of candidates correctly located the centre of mass of the pendulum bob.
Many of the wrong responses were at the point of attachment of the string to the support or
the bob.
(b)
Most candidates were able to calculate the period in (b)(i) and many correctly calculated
the number of oscillations per minute in (b)(ii).
(c)
The vast majority of candidates scored both marks. Of the remainder, some appeared to
not have access to a calculator so were unable to carry out the multiplication. Some
correctly wrote the substituted equation and then performed a division on the calculator.
(d)
Around half of the candidates gained the mark here. Many responses indicated that
candidates had not read the question properly. In the correct responses, candidates
successfully made the link that the moment increases due to the increase in the
perpendicular distance. Some candidates failed to gain the mark by not being comparative
in their answer.
E2.
(a)
(ii)
(b)
E3.
(i)
The calculation was well done with 92% of candidates achieving full marks.
Just over 85% of candidates were able to identify the correct units for a moment from
the list.
Not many candidates gained both marks for this question, mainly because they did not
give a full explanation. A significant number of candidates lost marks by using careless
terminology, with a considerable number of references to ‘centrifugal force’, ‘pressure’,
‘leverage’, and a few of ‘momentum’ and/or ‘weight’.
Foundation Tier
(a)
In this part the examiners were looking for a correct statement which would explain, or help
to explain, the term centre of mass. The candidate did not have to account for cases in
which the centre of mass lies outside of the object but, even so, appropriate responses,
for example ‘the object will balance if it is supported at its centre of mass’, were rare. ‘The
centre of the object’ and ‘the mass will be the same on both sides’ were fairly popular, but
incorrect, responses.
(b)
The specification states that ‘candidates should be able to describe how to find the centre
of mass of a thin sheet of material’. Some invalidated their responses either by clamping
the sheet or by suspending it through one hole and hanging the plumb line from the other
hole. Others ignored any sensible consideration of the size of the sheet and bent it so that
the 100 mm long rod could go through both holes. However, there was a minority of
candidates who seemed to understand what was required.
(c)
Only a minority of candidates were able to recollect the equation to calculate the turning
effect of a force. A large proportion injected the words mass and/or height into their
erroneous responses. Some were able to find the product of 44 and 0.4 but some spoilt
their efforts by increasing or decreasing by a factor of 10 or by doubling the distance to 0.8
m. If there was an attempt at a unit it was usually to suggest N/m.
Page 25 of 28
Higher Tier
(a)
In this part the examiners were looking for a correct statement which would explain, or help
to explain, the term centre of mass. The candidate did not have to account for cases in
which the centre of mass lies outside of the object but, even so, appropriate responses,
for example ‘the object will balance if it is supported at its centre of mass’, were not too
frequent. ‘The centre of the object’ and ‘the mass will be the same on both sides’ were
fairly popular, but incorrect, responses.
(b)
The Specification states that ‘candidates should be able to describe how to find the centre
of mass of a thin sheet of material’. Many were able to do this, and to express themselves
clearly, so gained all five marks. A small minority invalidated their responses by clamping
the sheet or by suspending it through one hole and hanging the plumb line from the other
hole. Some others ignored any sensible consideration of the size of the sheet and bent it
so that the 100 mm long rod could go through both holes.
(c)
Most candidates were able to recollect the equation to calculate the turning effect of a
force. Some injected the words mass and/or height into their erroneous responses. Most
candidates were able to find the product of 44 and 0.4 but some spoilt their efforts by
increasing or decreasing by a factor of 10 or by doubling the distance to 0.8 m. A
significant minority gave the unit as N/m. It could be that some correctly knew that their
answer was in newton-metres but incorrectly thought that this was the way to write its
symbol.
E4.
(a) For a question testing recall of a standard experiment given in the specification, the
responses produced were very disappointing. A few students gave a sufficient explanation
to score three marks. Just over a tenth of students did not attempt it at all and a further two
fifths of students did not score any marks. Most students did not describe the correct
experiment and of those who tried to do so the descriptions were often superficial and
inadequate. There was some confusion as to what is meant by a plumb line and a number
of students do not know the difference between vertical and horizontal. Some students
scored marks by drawing a good, labelled diagram.
(b)
E5.
This question was not attempted by a few students and a further half of students did not
score any marks. Many students incorrectly suggested repeating the experiment with
another piece of card or comparing their results with other groups.
(a) Many candidates were able to explain, either on the diagram or in writing, that the line
of action from the centre of mass falls vertically and is within the wheel base.
However, very few indicated that the weight of the tractor results in a moment which keeps
it turned to the slope or alternatively that if the line of action were to fall to the right of the
right wheel in the diagram the resulting moment would topple the tractor.
(b)
The great majority of candidates gained two marks because they knew that having a wider
wheel base and a lower centre of mass would increase the tractor’s stability.
Page 26 of 28
E6.
(a) Many candidates failed to gain marks here as their responses were definite ie it is
where all the mass is.
(b)
Just over half of the candidates were able to make two clear comparative statements with
a further quarter able to make one correct clear statement.
(c)
Many responses were not worth a mark because candidates had not used the terms as
instructed in the stem of the question. It was encouraging to note that the fuller answers
often used the phrase ‘resultant moment restores equilibrium’ or words to that effect.
E8.
(a)
(ii)
(i)
Nearly all students knew that the moment of a force is the turning effect of the
force.
Less than half of the students were able to state what is meant by centre of mass of
an object. Many referred to a region within the object rather than a point.
(b)
Almost all students were able to calculate a moment of a force.
(c)
(i)
Very few students scored the three marks for describing and explaining the
movement of a previously-balanced plank whose pivot had been moved away from
the centre of mass of the plank. The idea that the weight of the plank now provided a
moment was not understood.
(ii)
This high-demand calculation was successfully performed by about a quarter of the
students.
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Resource currently unavailable.
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