Although many students will be familiar with the definition of vector and scalar quantities others may have glossed over these ideas. Such students may feel nervous about vector quantities and will find simple mathematical analysis quite daunting at the outset. This is particularly noticeable if this topic is taught early in the course. It is important that students get a feel (literally!) for vectors alongside an understanding of their mathematical representation. It is worthwhile spending a little time on this introductory episode even if your specification does not make reference to it as it supports the work on moments and turning effects.

It is valuable to use practical work, computer simulations and pen-and-paper exercises to build student confidence in this topic.

Lesson Summary

• Discussion: Define scalar and vector quantities; examples (15 minutes)
• Presentation and discussion: Adding vectors at right angles (20 minutes)
• Student experiment: Displacement and vector addition (20 minutes)
• Worked examples: Adding perpendicular vectors (20 minutes)
• Student question: Practice for students (15 minutes)

Discussion: Define scalar and vector quantities; examples.

For some students this will be a revision session. Others may be less familiar with the concepts. Use question and answers to establish the knowledge of the group.

Start by defining the vectors as quantities with magnitude and direction. Compare these to scalar quantities (magnitude only). Use class discussion to develop a list of vector and scalar quantities (e.g. vectors: velocity, displacement, force; scalars: temperature, speed, distance, energy). It is useful to spend a little time on the difference between velocity and speed.

Discuss examples of simple, parallel vector addition. For example, a bullet fired from a moving train or a sled pulled across the snow against a constant frictional force. These easy examples build confidence in the basic principle. Extend this to anti-parallel vectors (i.e. vectors in opposite directions along same line; they subtract).

Discussion and demonstrations: Adding vectors at right angles

This leads to a consideration of non-parallel vectors. Begin with vectors at right angles.

Episode 201-1: Adding velocities (Word, 79 KB)

This demonstration takes a little time to set up and is over pretty quickly. However, it is worth the effort. You can perform the presentation with one student helper or you can act as a conductor of two student volunteers.

Make clear that the ink trail is the result of adding two perpendicular velocities.

Measure the duration of the journey to establish the three velocities.

Repeat the presentation with a clean sheet of paper. This time have the horizontal trolley moving at a new velocity to give a different gradient to the ink trail.

Draw a cleaned up version of the results on a board or ask the students to do this in their notes. The diagram should include a scale so that direct measurements can be taken. You will get something like this.

It can be seen that the magnitude of P can be calculated using Pythagoras’ theorem:

P² = v_horizontal² + v_vertical²

This can be checked against the results from the presentation. It can also be seen that a scale diagram can be used to predict the magnitude of P if the two horizontal and vertical components are known.

Simple trigonometry leads to the relationships v_horizontal = Pcos( θ ) and v_vertical = Pcos(φ)

This can also be checked against the results from the presentation. This is the resolution of the velocity P into horizontal and vertical components. Further discussion will show that:

v_horizontal = Pcos(φ) and v_vertical = Pcos( θ )

Students should note down these important relationships which will be used in the worked examples and student questions.

Student experiment: Displacement and vector addition

Episode 201-2: Displacement and vector addition (Word, 38 KB)

Your students can now put the theory to test, weather permitting! This activity begins with adding orthogonal vector displacements which will confirm the class discussions. The third activity sums displacements which are not at right angles. The simplest method of estimating the sum is to use a scale drawing – this may require explaining and extending to the general principle of the vector triangle. Students should be asked why Pythagoras cannot be applied to the third situation.

This should be considered as a quick activity to consolidate various ideas. Some students will believe that they know all about this; nevertheless, the practice is useful.

Make the addition of vectors by scale drawing explicit – a diagram such as the one below should be included in the discussion; in each case, the red arrow is the sum of the green and blue arrows.

Worked examples: Adding perpendicular vectors

Episode 201-3: Using the components of a vector (Word, 109 KB)

Use these examples to back up the work on vectors at right angles. The third part of the question is more difficult and shows the care needed in tackling such problems. The question is rather long and could be started in class as a discussion and left to finish for homework.

Student question: Practice for students

Episode 201-4: Flying in a side wind (Word, 39 KB)

This standard question will enable you to identify those students that have grasped the basic idea.

In this episode, students will learn about the conditions for static equilibrium (excluding moments of forces).

Lesson Summary

• Demonstration and discussion: The washing line (10 minutes)
• Student experiment: Forces in different directions (15 minutes)
• Student experiment: Forces in equilibrium (20 minutes)
• Worked example and Demonstration (15 minutes)
• Student activity: An estimation (15 minutes)
• Student questions: Two problems (30 minutes)

Discussion and demonstrations: The washing line

Take 5 m of good strong string. Ask two pupils to take either end and pull the string taut. Hang a 1 kg mass (10 N force) from the middle of the string. The pupils will find it very difficult (impossible, in fact) to keep the string horizontal.

Drawing a vector triangle of the forces will show why the pupils have had such problems.

Student should understand that the sum of the vertical components of the tensions in the string is equal in magnitude (and of opposite sign) to the weight hanging from the middle.

Similarly, the horizontal components of the tension in the strings are also equal and opposite. Why must this be the case? (Otherwise the string would start to move horizontally.)

You could finish the discussion with a statement of the principles of equilibrium. For an object to be in equilibrium the forces on it must be balanced; we can check this by resolving in two perpendicular directions. This will give the next activity a sense of a corroborative experiment.

Student experiment: Forces in different directions

This activity builds upon the opening discussion. Encourage the students to discuss the questions posed on the worksheet – if they don’t it just becomes a tug-of-war. Gather the class together at the end of the activity to compare ideas and clear up misconceptions.

Episode 202-1: Forces in different directions (Word, 35 KB)

Student experiment: Forces in equilibrium

This activity focuses on vector triangles. The fact that the triangles the students draw won’t quite close can be a springboard for discussion but you should not allow it to dominate the thinking of the less able students – the purpose of the demonstration is to give practice in using a graphical technique to consider forces in equilibrium, and show that it works!

Episode 202-2: Forces in equilibrium (Word, 39 KB)

Worked example and Demonstration

You can use this worked example in conjunction with the apparatus described in the question.

This example considers forces in equilibrium in a new situation – a trolley on a slope.

Having completed sections a – e, good students might like to resolve parallel to the runway to show that

Fcos( θ ) = Wsin( θ )

Episode 202-3: Trolley on a slope (Word, 27 KB)

Student activity: An estimation

Students can apply what they have learned to the equilibrium of a pylon supporting power lines. You may find it useful to give a lead in to the question and draw the diagram of the situation out for them. A picture can be used if you do not have pylons in the locality.

You may want to model the situation in class. In this case, a length of chain between two stools will show the students the effect and allow you to rehearse the calculations with them.

Episode 202-4: Forces acting on a power line (Word, 26 KB)

Student questions: Two problems

The first question should be a straightforward confidence-booster as it mirrors the algebraic analysis covered in the episode.

The second question is another scale drawing exercise. Mathematically fluent students should be encouraged to analyse the problem trigonometrically using the sine rule.

Episode 202-5: Questions on forces and equilibrium (Word, 56 KB)

Many students will recall the principle of moments from earlier work. The application of moments to situations beyond the simple see-saw examples met earlier in the school career can cause some students difficulty. It is important to ensure that each step in the argument leading to an application of moments to general equilibrium situations is made as clear as possible.

Lesson Summary

• Discussion: Moments and turning effects (10 minutes)
• Student experiment: Weighing a broom/laboratory stand (10 minutes)
• Student experiment: Centre of gravity of a student (5 minutes)
• Discussion: Conditions for equilibrium (5 minutes)
• Student experiment: Forces on a bridge (15 minutes)
• Worked example: The bookshelf (10 minutes)
• Student questions: Moments (15 minutes)

Discussion: Moments and turning effects

Your students will have come across these concepts already but their understanding may be shaky. It is therefore worthwhile to clearly define a moment and a couple thus:

It is also sensible to remind students of the equilibrium condition for moments (zero net moment about any point) before considering the first activity.

Student experiment: Weighing a broom/laboratory stand

Episode 203-1: Using the centre of gravity to find the mass of a broom (Word, 28 KB)

If you have insufficient brooms to go around you can use a lab stand with a weight of a few newtons hanging from it. (Take care that feet don’t get injured by dropping weights.) This has the advantage of allowing all the students to have a go at the experiment and allowing for some useful comparison of results. For example, did all the students get the same result? If not, why not? To what degree of precision should the result be quoted?

The students will need to know the importance of the centre of gravity. Once this is established the experiment becomes fairly simple, but a little more interesting than see-saws.

Student experiment: Centre of gravity of a student

Find the position of the centre of gravity of a student. This reverses the process of the preceding activity.

You need to be a little sensitive about who you choose for the student – you can always volunteer yourself. This activity can be adapted to work for Action Men and Barbie dolls!

Episode 203-2: Centre of gravity of a student (Word, 40 KB)

Discussion: Conditions for equilibrium

Careful questioning will encourage the students to formulate the conditions for equilibrium. They will readily state that the sum of the turning effects must be zero (or words to that effect such as: clockwise moments = anticlockwise moments) but they may need to be reminded that the resultant force must also be zero.

Student experiment: Forces on a bridge

This exercise gives practice in combining the two sets of equilibrium conditions. It can become as complex as required. The metre rule must be horizontal if possible (never that easy). There is a degree of practical difficulty which gives students an opportunity to develop practical skills, but some may need help.

The results will not match up particularly well (especially with lighter weights or weights far from the centre of gravity). Use this as an opportunity for discussion but emphasize the usefulness of the mathematical approach.

Episode 203-3: Forces on a bridge (Word, 74 KB)

Worked example: The bookshelf

Episode 203-4: Worked example (Word, 26 KB)

This is a standard example and links equilibrium conditions with resolved forces. This link is not obvious to all students – it is advisable to proceed with care! A non-mathematical class may lose confidence in the previous work if they are confused by this synoptic question across Episodes 1–3. Check carefully whether the specification you are following requires this before proceeding and choose further examples with care.

However, mathematically-inclined post-16 level students will enjoy such examples and should be encouraged to tackle them.

Student questions: Moments

Various situations requiring the ideas of moments are used. You may wish to set extra questions to more able students

Episode 203-5: Moments questions (Word, 146 KB)