Balancing 3
Mo has left, but Meg is still experimenting. Use the interactivity to help you find out how she can alter her pouch of marbles and still keep the two pouches balanced.
Problem
Mo has gone home now, but she left her pouch of marbles with Meg.
Meg has hung Mo's pouch at a fixed point on the right of the balance, and is experimenting with her pouch of marbles on the left.
Meg has discovered that if she puts 8 marbles in her pouch when it is 100cm from the centre, then it balances with Mo's pouch.
Use the interactivity to explore this further.
Full Screen Version
Fill in the table below to show how she can make it balance with different distances and different numbers of marbles.
|
Distance from
the centre
|
Number of
marbles
|
| 100cm | 8 |
| 50cm | |
| 25cm | |
| 40cm |
Explore this further.
Without using the interactivity, can you now work out how Meg would need to alter her pouch to balance it with Mo's?
|
Distance from
the centre
|
Number of
marbles
|
| 200cm | |
| 20cm | |
| 25 | |
| x cm | |
| x |
Try to explain how you worked it out.
Getting Started
Use the interactivity to help you complete the tables.
Look at your results in the table.
What happens if you double something on the left hand side?
What if you treble it, or halve it?
How many marbles would Meg need if her pouch was 1cm from the centre?
Where would Meg have to hang her pouch if it contained only 1 marble?
Student Solutions
Well done to Matthias and Aneesh of the Lady Barn House School who made some interesting observations.
If the distance is doubled you need to halve the number of marbles. If the distance is reduced by a factor of 10 you must multiply the number of marbles by 10.
The numbers are also interchangeable so if the distance from the centre is 50cm you will need 16 marbles to balance the scales, and if you are 16cm from the centre you will need 50 marbles to balance the scales.
Congratulations to Jake of Horris Hill who gave a very full answer to this problem.
In his solution, Jake used the term "effective weight" to explain the measure given by weight x distance from the centre, but we have replaced this with the word "moment" which is more conventionally used.
The particular problem with this puzzle is that the bag can be any weight and any distance from the centre, making two factors to consider. However, after experimenting with the scales, I found there was a relation between the weight and distance of one bag and that of the other bag when the scales balanced.
I started to work with the numbers in my head, and I found that the relation was between the products of the two pairs of numbers. In other words: weight of Meg's bag x distance from centre of Meg's bag = weight of Mo's bag x distance from centre of Mo's bag.
Looking at a first class lever (e.g. a see-saw), you can see that the further away the effort (Meg's bag in this case) is from the fulcrum (i.e. middle of the scales), the less force (weight in this case) is needed to lift the load (Mo's bag in this case). This means the bigger the distance and the heavier the bag, the more weight can be lifted, which helps to create a unit of measure that we can compare instead of two separate units.
We call this the moment.
If two bags have the same moment, they will balance. To make it clear what the moment of a bag is, this is the formula:
moment = weight x distance from centre.
Going back to the problem, the unit of the moment is marbles (weight) x cm (distance from middle of scales), and this arms us with a solution to the problem. A quick glance at Mo's bag tells us that the moment is 800 (marbles x cm), meaning that any bag with a moment of 800 (marbles x cm) will balance with it. Once you know this, all you need is one of the measurements
| Distance from the Centre | Number of Marbles |
| 100cm | 8 |
| 50cm | 16 |
| 25cm | 32 |
| 40cm | 20 |
| Distance from the Centre | Number of Marbles |
| 200cm | 4 |
| 20cm | 40 |
| 32cm | 25 |
| x | 800/x |
| 800/x | x |
Incidentally, the highest number of marbles possible on the interactivity is 32, meaning the nearest possible distance to the centre whilst keeping the bags balanced is 25 cm (see chart, above).
Teachers' Resources
This problem offers an interactive environment that introduces students to moments of force by asking them to consider how a balance is affected by altering the weights and the distances from the centre.
This problem is a follow up to Balancing 1 and Balancing 2.