Starters and activities in Mechanics
MEI conference 2012
Keele University
Starters
Centre of mass: two counter-intuitive stable positions of
equilibrium
The directions of displacement, velocity and acceleration
The total surface force on an accelerating car
Relative velocity demonstrations and a simulation in GeoGebra
Drawing diagrams in mechanics
Interpretation of a velocity-time graph
Activities
An experiment and mathematical model for sliding
An experiment and simulation of three forces in equilibrium
Simulations to aid understanding of projectile motion
A simulation of forces on a box causing sliding or tipping
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Starters and activities introduction
1.
Emily Rae and David Holland
2.
These starters and activities are intended to
suggest types of things you can do for yourselves
to meet the needs of your students.
3.
Students have a ready interest in Mechanics
because of the way it ‘explains’ things. This
intellectual curiosity is a much better motivator
than appeals to utility.
4.
Some of Mechanics is quite hard. Students do
not give up using their own limited or wrong
models just because we give them better ones.
They have to be confronted with situations that
evidently cannot be correctly analysed by their
limited or wrong models.
5.
An underlying theme will be the use of
simulations as a powerful learning aid. This is a
type of aid that has vastly increased in
importance with the development of computers.
Centre of mass – discussion starters
The first activity is a common party trick which is intended to address the issue of whether
an object’s centre of mass can exist outside of the object itself (for example, in an ‘L’
shaped object) – a concept which students can struggle with when calculating the position
of a centre of mass.
First, position a fork and spoon end-to-end by pushing the curved part of the spoon
through the prongs of the fork. Balance the conjoined cutlery on the end of a matchstick
(see image) then balance the other end of the match onto the rim of a glass.
The whole system looks improbable, as though it should not be in equilibrium. Ask the
students why they think it works, and why the fork and spoon do not fall off the end of the
matchstick (the reason is that the centre of mass of the spoon-fork-matchstick system is in
a vertical line through the point of contact of the matchstick with the glass)
Another discussion starter on the topic of centres of mass is to explore where the centre of
mass will be in a can of liquid at various stages of the can emptying. When the can is
empty or full, the centre of mass will be in the same place (roughly halfway up the can)
However, when a small amount of liquid is removed from the full can the centre of mass is
lowered.
Ask the students when they think the centre of mass will be at its highest, and when it will
be at its lowest.
It is also possible to drink just the right amount of liquid to make the can balance on its rim
without falling – when the can is balanced the centre of mass is located directly above the
point where the rim sits on the surface of the table.
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The direction of displacement, velocity and acceleration
Notes for teachers
Students often make mistakes when asked for the direction of displacements,
velocities and accelerations. Common errors are to confuse the direction of motion
with the direction of the displacement and, particularly, assume that the direction of
the acceleration is the direction of the velocity.
Getting a class to consider some sort of periodic motion is very effective. Useful
scenarios are
walking up and down a line in front of the class,
considering a vertical mass-spring oscillator (with a long period),
considering the motion of a lift.
In each case an origin and the positive direction is defined and for different stages of
the motion the class have to decide on the sign of the displacement, velocity and
acceleration. It is a good idea to use the ‘walking’ and ‘lift’ examples initially as they
can be ‘paused’ to allow discussion.
After a longer introductory session, the exercise can be repeated quickly as a starter to
reinforce the ideas. Repeating one of the scenarios with a different origin and/or
positive direction is very effective.
Taking the lift example, one could discuss the non- stop motion from the ground to
the 2nd floor and then the non-stop return journey. Put up a diagram showing the
origin at 1st floor level and the positive direction upwards. Suppose that the lift is
initially at ground floor level.
Setting off up from ground floor
Passing 1st floor
Approaching 2nd floor
Setting off down from 2nd floor
Passing 1st floor
Approaching ground floor
displ –ve
displ 0
displ +ve
displ +ve
displ 0
displ –ve
vel +ve
vel +ve
vel +ve
vel –ve
vel –ve
vel –ve
acc +ve
acc 0 (discuss?)
acc –ve
acc –ve
acc 0 (discuss?)
acc +ve
Now try this with positive downwards. Now try with the lift stopping at 1st floor
level. Now try…
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Total surface force on a car
Each of the diagrams shows the total surface contact forces acting on the front and on
the rear wheels of a car on a straight horizontal road. Air resistance should be
neglected.
For each case, suggest a scenario and the motion that could be taking place.
Notes for teachers
Students may not be familiar with the use of ‘total surface contact force’ and this may
require some discussion. Perhaps a further discussion might be about why we usually
find it more convenient to deal with this force in component form, i.e. as friction and
normal reaction.
Many students will not be familiar with the fact that the frictional force on a car acts
towards the front when it is accelerating forwards and this often provokes discussion.
In a real situation, the two normal reaction forces will change under acceleration and
deceleration while retaining the same total magnitude (equal to the magnitude of the
weight of the car). Some students may be aware of this (the change in load on the
front and rear suspension is indicated by the car’s nose rising under acceleration and
dropping under braking). This is complicated stuff that is probably better mentioned
only briefly but the normal reactions in the diagrams have been deliberately given
different values in the different situations.
Some possible scenarios are:
1
At rest or constant velocity
2
Accelerating with rear-wheel drive
3
Accelerating with front-wheel drive
4
Accelerating with 4-wheel drive
5
Decelerating under braking (or 4-wheel drive accelerating backwards)
6
Accelerating forwards with the rear handbrake on
Draw the picture if the car has front-wheel drive and is accelerating backwards with
the rear handbrake on.
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Relative velocity
In each case the train is moving at 0.2ms-1, relative to the table top. If the table is also
moving with a constant velocity of 0.1ms-1, what is the resultant velocity of the train? In the
last two cases, draw a velocity triangle and calculate the magnitude and direction of the
resultant velocity.
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Relative velocity
Notes
The relative velocity activity is designed to help students understand the wording of
situations such as “The velocity of a boat, relative to the water, is…” In this case, the train’s
velocity “relative to the tray” is 0.2ms-1, while the tray also moves with its own velocity. The
resultant velocity can be found using a vector triangle, allowing students to practise this skill.
As a visual aid, a toy train may be used with this activity. Ask students to estimate the
velocity of the train (the one I used travelled approximately 20cm in 1 second, so I used
0.2ms-1 throughout the example). Put the train on a large tray (or table top), and when the
train is moving in a straight line ask a series of questions such as “What would the resultant
velocity of the train be if I moved the tray at 0.1ms-1 in the same direction as the train?”
The idea of relative velocity can also be demonstrated dynamically (for instance using
Geogebra), with a velocity triangle showing two vectors (in the case below, u and v) and a
third vector (w) showing the relative velocity of v relative to u.
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Drawing diagrams worksheet
Draw diagrams for the following scenarios including all the forces acting with arrows
and labels. Indicate any force whose direction may not be known.
1
A heavy box on a smooth slope.
2
A heavy box on a rough slope.
3
A heavy box on a rough slope being held in equilibrium by a string parallel to
the slope.
4
A heavy box on a rough slope being pulled up the slope by a string parallel to
the slope.
5
A heavy uniform ladder inclined at 20° to the vertical standing on a rough
horizontal floor and resting against a smooth vertical wall.
6
As in 5 but with a rough vertical wall.
C
A
B
D
80 N
The diagram shows a box of weight 80 N. BD is a light string in tension. The system
is in equilibrium
Draw diagrams for the following cases.
7
AC is a light string that passes through a small smooth ring attached to the box
at B.
8
AB and BC are light strings, each tied to the box at B.
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Drawing diagrams worksheet
Draw diagrams for the following scenarios including all the forces acting with arrows
and labels. Indicate any force whose direction may not be known.
In the answers suggested: W is weight; R, R1, S etc are normal reactions; T, T1 etc are
tensions; F, F1 etc are frictional forces.
1
R
A heavy box on a smooth slope.
W
R
2
F
A heavy box on a rough slope.
W
3
A heavy box on a rough slope being held in equilibrium by a string parallel to
the slope.
R
T
F
W
direction of F not known
There is a lot to discuss in this example. The box may not be in limiting equilibrium
and the direction of F depends on the relative magnitudes of T and of the component
of W down the slope.
4
A heavy box on a rough slope being pulled up the slope by a string parallel to
the slope.
R
T
F
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5
A heavy uniform ladder inclined at 20° to the vertical standing on a rough
horizontal floor and resting against a smooth vertical wall.
6
As in 5 but with a rough vertical wall.
F2
S
S
20°
20°
R
W
R
W
F1
F1
Q6
Q5
The diagram shows a box of weight 80 N. BD is a light string in tension. The system
is in equilibrium
C
A
B
D
80 N
Draw diagrams for the following cases.
7
AC is a light string that passes through a small smooth ring attached to the box
at B.
8
AB and BC are light strings, each tied to the box at B.
TN
A
TN
B
C
T2 N
A
T1 N
T3 N
B
T4 N
D
D
80 N
80 N
Q8
Q7
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A kinematics example
v m s -1
velocity
5
0
5
time
ts
1
Draw the speed-time graph corresponding to the given velocity-time graph.
With a suitable label for your graph, use the velocity axis for speed.
2
Find the areas between the velocity-time graph and the time axis. Interpret
these as displacements.
3
The velocity-time graph describes the motion of an object in the interval
0  t  10 . Initially ( i.e. when t = 0) the object is at A. The +ve direction is
vertically upwards. For this object find
(i)
(ii)
(iii)
(iv)
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the direction of its motion when t = 2 and when t = 7,
the distance it travels in the interval 0  t  10 ,
its greatest displacement from A in the interval 0  t  10 ,
the closest the object gets to A in the interval 4  t  10 and when this
occurs.
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velocity
speed
0
time
2
Areas are 0.5 below, 6 above, 2.5 below, 0.5 above so displacements are
0  t 1
0.5 m vertically downwards
1 t  5
6 m vertically upwards
5t 9
2.5 m vertically downwards
9  t  10
0.5 m vertically upwards
3
(i)
(ii)
(iii)
(iv)
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At t = 2 +ve v so vertically upwards.
At t = 7 -ve v so vertically downwards
0.5  6  2.5  0.5  9.5 m
– 0.5 + 6 = 5.5 m vertically upwards (after 5 seconds)
At t = 4 it is – 0.5 + 5.25 = 4.75 from A. Up to t = 5 it is getting
further away ( by 0.75 m). From t = 5 to t = 9 it is getting closer ( by
2.5 m) and after t = 9 it is getting further away again. So closest is
4.75 + 0.75 – 2.5 = 3 m and this occurs when t = 9.
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Sliding across the floor
For an object sliding across a horizontal floor, investigate
the relationship between the time it takes the object to
come to rest and the distance it slides in that time.
You should construct a mathematical model for the
situation, devise an experiment to make direct
measurements and then compare the experimental results
with the prediction of the model.
You have a stop watch, a measuring tape and an object.
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Notes for teachers
Assumptions:
floor uniformly horizontal
floor uniformly rough
object doesn’t bounce
object doesn’t spin
no air resistance
….
General matters
It is not possible to improve accuracy by repeating the experiment
exactly.
Could we scale this down and do it with coins on a table?
The model
RN
direction
of motion
m kg
a m s -2
FN
mg N
According to the assumptions, F is constant and there are no other forces.
Applying N2L in the direction of motion
F
– F = ma so a   .
m
As F and M are constant, a is constant and we may use the constant
acceleration formulae.
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We would find it hard to find the initial speed of the object but do know it
comes to rest so the suvat formula we need should involve s, v, a and t.
1
F 2
1 F 
Using s  vt  at 2 gives s  0  t     t 2 so s 
t
2m
2
2 m 
This means that a plot of s against t 2 should give a straight line through
the origin.
A refinement would be to find an expression for F in terms of the
coefficient of friction using F   R . Could we measure ? Could this
be a way of measuring ?
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Three forces in equilibrium
Two pulleys are at the same height and in the same vertical plane. A
string has objects P and Q with masses m1 and m2 attached to the two
ends. It passes over both pulleys and a third object R, of mass m3, is
attached to the point X of the string between the pulleys.
 
X
R
m1
m3
P
Q
m2
With the system in equilibrium, the angles the two parts of the string
attached to the central object make with the vertical are measured for
various values of m1, m2 and m3.
Form a mathematical model for this situation and compare the predictions
of the angles with the values measured.
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Some questions that could be asked of students
What assumptions are needed for a simple mathematical model and what
are their relative importance?
What measurements are necessary to be able to compare the predictions
of the model with the experimental results?
What problems do you think there would be in making sufficiently
accurate measurements? Could you avoid the use of a protractor?
Some notes for teachers
Assumptions
pulleys smooth
strings light
…..
Need the string be inextensible?
Difficulties with the measurements
There is some difficulty in measuring the angles using a protractor –
students might find it easier to measure lengths and then use trig.
Model
T1
 
T2
X
T1
m1
P
m1g
T3
T3
R
m3
m3g
T2
Q
m2
m2g
In the diagram take all the units to be SI
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Solution using the triangle of forces and the sine rule
forces at X

T1

T2
T3
Using the sine rule
T1
T
T3
T3
etc
 2 

sin  sin a sin(180  (   )) sin(   )
Of course, the problem may also be solved using resolution.
Possible learning outcomes:
 Each different equilibrium position is associated with a
particular position of the objects and there are limitations on the
values of the masses.
 There are many other possibilities
General matters
How much help would your students need with creating the model?
What the relative advantages and disadvantages of using a simulation
instead of this experiment?
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Why use simulations as well as or instead of experiments?
What do we get from experiments?
They allow us to consider the approximations to ideal behaviour by
making assumptions such as lightness, smoothness, rigidity,
inextensibility, the value of g and the suitability of Coulomb’s law,
Hooke’s law etc.
Students who have prior practical experience of a situation are more
likely to recognise it and less likely to accept inappropriate answers
Students have rich experience of mechanical situations e.g. cars
(including learning to drive), lifts, boats, fun-fairs, watch F1 …
Experiments don’t just provide new experiences they also help them
interpret the ones they have already had.
There can, however, be problems with using experiments to introduce
students to ‘standard’ situations. Experiments
 may be influenced by large but ‘hidden’ factors,
 do not necessarily make it easy to explore a wide range of conditions,
 are sometimes very time consuming.
What do we get from using simulations?
Simulations allow students to familiarise themselves with the predictions
of a mathematical model
 quickly
 over a wide range of controlled parameter changes
 interactively
Projectile motion
Experiments with projectiles to demonstrate non-resisted motion are very
useful but suffer from
 the difficulty in reproducing starting conditions,
 the difficulty in seeing what is happening – a projectile with slower
motion usually has large air resistance
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Some helpful simulations of projectiles moving with negligible air
resistance
1. A projectile moving slowly on its trajectory with different angles and
speeds of projection.
2. Complete trajectories and how they vary for different angles and
speeds of projection.
3. The constancy of the horizontal component of velocity of a projectile
and the changing nature of the vertical component.
4. The family of trajectories with the same initial speed but different
angles of projection. This shows the envelope which is the ‘parabola
of safety’ and that there are two trajectories that pass through each
point inside this envelope.
5. Two different trajectories with the same time of projection and the
same initial speed. One can compare the progress of the projectile on
the two trajectories.
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