MEI Conference 2016

Preparing to teach moments

Simon Clay

simon.clay@mei.org.uk

mailto:simon.clay@mei.org.uk

2 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0

Is it possible for this to happen? Teacher notes

From Twitter via @MaxCRoser

Points to consider:

You might want to encourage students to consider:

- which areas of mechanics are applicable in this situation

- any measurements they require such as the mass of car, mass of ‘average’ person, dimensions of

car, position of centre of mass of car, etc

- any assumptions they need to make to model the situation

Extension:

The report suggests that the car was moving when the people jumped onto the bonnet. Model this

situation. Does this change your conclusions in any way?

3 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0

Is it possible for this to happen? Student sheet

From Twitter via @MaxCRoser

Extension:

The report suggests that the car was moving when the people jumped onto the bonnet. Model this

situation. Does this change your conclusions in any way?

Task:

Decide what would need to happen in order for the above situation to be possible.

Make sure that you can justify your model and any measurements or assumptions

you have used.

4 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0

Designing a ‘Teeter-Totter’

Teacher notes

This task is aimed at reinforcing students’ knowledge of the theory of moments and help to develop

their ability to apply it. It is designed to be open-ended with plenty of scope for allowing students to

generate a suitable solution. There is deliberately very little guidance on the student sheet below as

this places students in a position where they have to make decisions and identify assumptions they

are making.

Students could be given the prompt below as a handout or simply shown the photo with a class

discussion to help direct their initial thinking. However, be aware that this can stymie creative

solutions!

The apparatus is called a ‘Teeter-Totter’ although this could well not be its ‘official’ name!

Encourage students to give as much detail as possible in their design and justify any decisions they

make mathematically. Some prompt questions you may wish to use during the activity could be:

- What do you want the outcome to be for a person using this equipment?

- Who is the apparatus suitable for?

- How does the equipment behave for a small child? Teenager? Adult? Does this matter?

- How far does a child with a mass of 25 𝑘𝑔 have to walk along the beam before it pivots?

What about a person with mass 10 𝑠𝑡𝑜𝑛𝑒?

- What type of wood should be used? Why?

- What assumptions have you made in your model?

- What are the dimensions of the ‘Teeter-Totter’?

5 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0

Designing a ‘Teeter-Totter’

The photo below shows a piece of apparatus from a ‘Commando-style’ obstacle course, called here a

‘Teeter-Totter’. It consists of a log that is hinged on a pivot so as you walk up the beam from the

ground it gets to a point where it then tips and you can walk off the other end. (Note that you can see

a second one in the background.)

Task:

Produce a suitable design for a ‘Teeter-Totter’ justifying the decisions

you have made.

Indicate any measurements that would be required to produce your

‘Teeter-Totter’ and identify any modelling assumptions you have

made.

6 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0

Investigating Moments - Teacher notes

Equipment: - a selection of metre rulers

- selection of masses, including 10 𝑔 masses totalling 200 𝑔

- Blu-tac

- Play-Doh

- Optional: some lengths of string for attaching rulers together

These investigations work well with students working in pairs. The above equipment needs to be

available for each pair. Following the initial experiment there are a series of prompt questions to

encourage students to think more deeply about what they have discovered. Students should also be

encouraged to ask their own questions and explore these.

Practical notes:

- A student can use their finger to balance the ruler but for increased accuracy they may wish to use

some sort of wedge or block instead. An effective alternative is to use the edge of a table.

- Masses can be attached to the rulers by using Blu-tac.

- Rulers can be attached to each through the use of Blu-tac or by tying a piece of string around them.

Investigation 1: Balancing a ruler

This investigation could be used with students who are yet to meet the theory of moments and be a

way in to exploring and generating the theory. It provides an opportunity for students to think through

the assumptions they make during the modelling process (for example, the beam is uniform, the mass

is a particle, etc.) and also confront some of the issues of experimental approaches such as the

accuracy of measurements.

You may need to encourage students to try several numerical cases in order to build up a data set.

Plotting a graph of distance of support from the end of the beam, 𝑑, against the total of the masses

supported by the beam, 𝑚, leads to some interesting models being produced.

Investigation 2: Beam balance

This investigation is split into two complementary parts. The first provides students with an

opportunity to apply their knowledge of the theory of moments and explore various solutions for a

specific problem as well as a solution for general masses. The second part offers the opportunity for

students to begin to use the theory to problem solve.

Adapted from activities of the same name in ‘Mechanics in Action’ (1990)

7 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0

Investigating Moments - Student sheet

Investigation 1: Balancing a ruler

Equipment: - a number of metre rulers

- at least five 10 𝑔 masses

- Blu-tac

1. A ruler will usually balance at its midpoint. Check that this is true for your ruler.

2. Attach a 10 𝑔 mass to the end of the ruler. Where does the ruler balance now?

Q: Can you predict where the ruler will balance when 20 𝑔 is placed on the end of it? What about

when using 30 𝑔? Continue to investigate until you have a conjecture that you can test with a

prediction.

Q: Can you explain what is happening?

Some further points to explore:

What happens when you have a number of rulers stuck together?

What happens when you place masses at various points along the length of the ruler?

d

8 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0

Investigation 2: Beam balance

Equipment: - a metre rule

- 40 𝑔, 70 𝑔 and 90 𝑔 masses

- Blu-tac

- a small and a large lump of Play-Doh or similar

Part A:

Place the 40 𝑔, 70 𝑔 and 90 𝑔 on a beam (metre ruler)

Q: Can you do this and make the beam balance?

Q: Is there more than one way of doing this? How many ways can you find?

Q: What about if the masses were 𝑥, 𝑦 and 𝑧?

Part B:

Using five 10 𝑔 masses and the beam can you find the masses of the small and large lumps of Play-

Doh?

Q: How large a mass can you measure?

Q: How small a mass can you measure?

Q: How accurate can you be in these measurements?

Some further points to explore:

Q: Can you find the mass of the beam using the equipment above?

Q: Can you place two 10 𝑔 masses on adjacent numbers on one side of the metre rule and then

balance these using three 10 𝑔 masses on the other side?

Preparing to teach Moments

Simon Clay

simon.clay@mei.org.uk

mailto:simon.clay@mei.org.uk

Session description This session is one of four designed to help teachers prepare for teaching

mechanics topics in the new A level from 2017.

"Preparing to teach moments" will cover some of the basic subject content,

links to GCSE and other A level topics, as well as exploring ideas and

approaches for teaching moments. The session will demonstrate how

simple practical classroom activities can provide a stimulus for students to

develop their understanding of moments. We will also consider how we can

develop models of situations involving moments.

The session is suitable for teachers who have not previously taught any

mechanics. It will also be suitable for teachers who have previously taught

M1 modules, but for whom moments has not been included within their

current M1 specification.

What will happen and why?

The particle model When a system of forces acts on a particle, the particle may be in one of the following states

- static equilibrium

- constant velocity

- accelerating in the direction of the resultant force

Typically in early study of Mechanics students have seen situations where bodies may be successfully modelled by single particles:

- a book at rest on a slope

- a book sliding down a slope

- a car and trailer accelerating along a straight road

Moving on from the particle model

The table could slide or topple

Maximising turning effect of a force

q

F

d

A simple experiment with your students

The moment of a force

O d

F

The turning effect of a force is called the moment of a force.

The moment of a force F about an axis through O

perpendicular to the plane containing O and the line of

action of F is F x d, where d is the perpendicular distance

from O to the line of action of F.

- Moment has sense, usually

described as clockwise or anti-

clockwise and is signed positive or

negative according to the

convention adopted for that problem.

- The SI unit of moment is the N m.

‘Weighing’ your piece of wood

Equipment: A piece of wood

A 100 g mass

A ruler

Task: Calculate the mass of the piece of wood

What examples of moments can

you think of?

Alternative bathroom scales?

Reformed A level Content

www.gov.uk/government/publications/gce-as-

and-a-level-mathematics

https://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematics

Simplifications

At A level, the following simplifications are made:

• Only static equilibrium situations are considered

• The moments are easy to calculate, mostly with forces

that are parallel

• We do not usually mention that the moment is about an

axis perpendicular to the plane through a point, say A,

we just say the moment about A.

Modelling assumptions

• The body is rigid - it is not deformed by the forces that

act on it

• The axis is fixed

• The body is free to rotate about the axis - for instance,

there is no frictional or other force impeding rotation

about a hinge

• The moment of the whole weight may be found by taking

its line of action to be through the centre of mass

Conditions for static equilibrium

For the equilibrium of a body it is necessary and sufficient

that the resultant of all the external forces is zero and that

the total moment of these forces is zero about any axis.

In the case of coplanar forces it is sufficient for the

equilibrium of a body that the resultant of all the external

forces is zero and that any point can be found about which

these forces have zero moment.

Solving problems

To meet the conditions for equilibrium we have to resolve in two directions (we choose) to establish a zero resultant force and take moments about one point (we choose) to establish that the forces have zero moment about this point.

As you might suppose, a wise choice of the directions and point can make the solution easier.

Not every problem requires two resolutions and moments taken

AQA A level Mathematics (2017)

draft sample paper 2, Q12

AQA A level Mathematics (2017)

draft sample paper 2, Q12

Calculating moments

But finding the perpendicular distance onto the line of

action of a force is not always quite so straightforward!

Example: Suppose that the force of 10 N acts through P

which is 4 m from O and that the force is at an angle of 60°

with OP.

O 4 m

P

60°

10 N

Calculating moments

10 cos 60 N O

4 m

P

10 N

60°

10 sin 60 N

Experiments - Investigating

moments

http://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.html

Interactive simulation

http://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.htmlhttp://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.htmlhttp://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.htmlhttp://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.htmlhttp://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.html

Experiments - The Ladder

‘Teeter-totter’ task Produce a suitable design for a ‘Teeter-Totter’ justifying the decisions

you have made.

Indicate any measurements that would be required to produce your

‘Teeter-Totter’ and identify any modelling assumptions you have made.

The car & the quayside

Via Twitter

Problem: Two supports

OCR A level Mathematics (2017)

draft sample paper 1, Q15

OCR A level Mathematics (2017)

draft sample paper 1, Q15

‘Rowing and the Same-Sum Problem Have Their Moments’

by John D. Barrow

Solving the boat ‘wiggle’

problem

http://arxiv.org/pdf/0911.3551v3.pdf

Rowing moments

http://arxiv.org/pdf/0911.3551v3.pdf

About MEI

• Registered charity committed to improving

mathematics education

• Independent UK curriculum development body

• We offer continuing professional development

courses, provide specialist tuition for students

and work with industry to enhance mathematical

skills in the workplace

• We also pioneer the development of innovative

teaching and learning resources

SessionH_Moments_Delegate_handout.pdf (p.1-8)SessionH_PreparingToTeachMoments_webversion.pdf (p.9-39)