Lesson video

Hello there, I'm Mr. Forbes and welcome to this lesson from the Forces Make Things Change Unit.

This lesson's all about balancing and in it you're gonna carry out an investigation to see what factors affect the balance of a beam and to do calculations involving balance.

By the end of this lesson, you'll have carried out an investigation into the factors that affect how a beam balances.

For that you're going to use a ruler and some masses.

You'd have discovered something called the principle of moments and then used that in a range of calculations.

And these are the keywords that'll help you during the lesson.

The first is equilibrium and an object is in equilibrium when there are no resultant forces on it and no resultant moments.

A moment is a turning effect of a force and we measure that in Newton meters.

And the principle of moments is a statement that states when the clockwise moments acting on an object are equal to the anticlockwise moments, the object will be balanced.

You can return to this slide at any point in the lesson.

The lesson's in just two parts and the first part you're going to carry out an investigation that will allow you to discover the principle of moments using a balanced beam.

And in the second part of the lesson you'll be using that principle of moment to solve a range of problems involving balance.

Let's get started with planning that investigation.

You will have studied moments in previous lessons, but as a recap, the moment caused by a force is given by this equation moment is force times distance, normal to the direction of the force or right angles to the direction of the force.

In symbols we can write that as M equals F times d, and where M is the moment measured in Newton meters, F is the force measured in Newtons, and d is the distance in the direction of the force and that's measured in meters.

Let's try an example to see if you can remember how to calculate moments.

So I'm going to calculate the moment caused when a force of 5.

0 Newtons acts 1.

5 meters from a pivot.

And to do that, I write out the equation moment equals four times distance.

I substitute in the two values from the question, the force is five Newtons, the distance 1.

5 meters and that gives me a moment of 7.

5 and I need a unit for that and that's Newton meters.

Okay, let's see if you can calculate a moment.

I'd like you to calculate the moment for the data in this question please.

So pause the video, do that calculation and restart.

Welcome back.

Well, you should have written the equation, substituted the two values in like this, 25 Newtons and 1.

2 meters and that gives you a moment of 30 Newton meters.

Well done if you got that.

So in this lesson, we're going to try and find out what factors affect the balancing of a beam and we're going to use a meter rule for that beam on top of a triangular wedge, something like this.

So we've got a meter rule there and it's on top of a pivot, which is a triangular piece of wood or plastic, something like that just on top of a flat desk.

The wedge is placed halfway along the length of the ruler so that the ruler balances.

So we've got the weight of the ruler acting down and there's going to be a reaction force from the pivot acting up right at the center and that will give a balanced rule.

You might find that your ruler doesn't balance perfectly in the middle, so you'll have to find that balanced point for yourself sometimes.

You'll also find that it's easier to balance the ruler if the pivot's not a perfect triangle.

If it's slightly flattened at the top, you'll find a simpler balance point.

So, try and find a pivot that's not perfectly pointy at the top.

If you placed a small mass on one side of the meter rule, then the meter rule rotate until it touches the desk and it'll stop rotating then.

So your guess situation like this where the weight of that mass is acting downwards on the ruler, causing it to rotate in this case clockwise.

So the clockwise moment produced by that mass is causing that rotation.

You can get the rulers to balance again by placing an identical mass the same distance from the pivot on the other side of it.

So a situation like this where I've got two masses, I've placed some 30 centimeters from the pivot, they're identical masses producing identical downward forces and that the beam balances again, the beam isn't accelerating and there's no resultant force acting on it and it's not starting to turn in any way.

The wedge in the center, that pivot must be causing an upwards force on it that counteracts the downward force of those two masses and the weight of the ruler.

So, I've got a situation where the reaction force is equal to all of those three down forces added together, the weight of the two masses and the weight of the ruler.

Now it's time for the first check for the lesson.

A balanced rule has a weight of two Newtons.

Two masses of weight one Newton are placed on it, as shown in that diagram.

I want to know what is the size of the upward force R.

So pause the video, make your selection from the list there on the left and restart please.

Welcome back.

Hopefully you chose four Newtons.

There are four Newtons downwards.

We've got one Newton and two Newtons and another Newton and that gives an upward force of four Newtons as well if the ruler is balanced.

So well done if you got that.

You can also balance the ruler when you place a single mass on one side and two masses on the other side of the ruler.

For that to happen, the single mass needs to be placed further from the pivot than the two masses on the other side.

So you can see in this situation, I've got a mass on the right-hand side, it's got a weight of W and it's 40 centimeters from the pivot.

And to balance that I can place two masses, so that gives a weight of two W on the left-hand side, but that's only 20 centimeters away from the pivot.

The ruler can also be balanced when the two masses are not placed on top of each other.

You have a single mass here on one side with the weight of W and the other mass is on the other side, two W's but at different distances.

Both of those masses on the left are closer to the pivot than the one on the right in this situation.

I've got my distances marked over here and again that rule will balance.

What you are going to do is carry out an investigation to try and find out different situations where the ruler balances.

So you're going to place masses on either side of the ruler and see if you can get positions where those balance against each other and the ruler doesn't start tilting to each side.

You use different combinations of masses and you're placing them on opposite sides of the ruler at different distances.

Some of the combinations that you'll find will balance and most will not.

So I've got situations like this, this situation balances if I position one weight 40 centimeters on one side and two weights 20 centimeters on the other.

But this situation doesn't balance.

So, you're basically trying to find out different positions where they will balance and rule out one's where it won't.

It can be difficult to get the ruler to balance perfectly, so you're going to have to judge a little bit whether or not you think the ruler is balanced.

If it's starting to tilt very slightly, you might judge that that's approximately balanced.

Okay, before you start the investigation, I'd like you to try and think about this situation.

I've got Andeep and he's trying to balance a ruler shown in the diagram here.

We've got three masses on it.

And as you can see it's not balanced, it's tilted to the right side here.

Which of the following changes do you think would cause the beam to balance?

And I've got three possibilities there.

So pause the video, read through those, and make your selections, and then restart please.

Welcome back.

Well, you should have chosen these two.

Move mass X further from the pivot to the left or move mass Z closer to the pivot on the right.

And the reason that that will cause it to balance is in the first situation, that's going to increase the anticlockwise moments to cause a bit more tilting to the left side there.

And in the second one you're gonna be decreasing the clockwise moments and both of those modifications would make that ruler balanced a bit better.

So well done if you selected those.

And now it's time for you to carry out the investigation.

So what you're going to do is set up the fairly simple apparatus like this.

You've got a wedge or pivot there resting on a desk and you've got balanced ruler on top of it and you've got a selection of masses to move to different positions along the left-hand side and the right-hand side of the ruler.

So, balance the meter rule with its center on top of that triangle wedge, carefully position different combinations of up to five 0.

5 Newton weights.

You don't wanna have too many weight or masses on there.

And adjust their positions until you get the ruler of balanced.

Record any combinations of masses and distances which cause the ruler to balance.

You don't need to record all of the possibilities that do not cause it to balance.

I want you to repeat that process until you find at least five different balancing combinations.

You're going to record your results in a table like this.

I've already recorded one set of results for you there but I'd like you to find at least five combinations where you'll get the ruler to balance with different masses on different sizes, sides, sorry.

So, when you're ready, carry out your experiment and then restart the video.

Welcome back.

Well, your results should look something like this.

I've got a completed table here with five different combinations of masses at different distances and for that, the ruler balances for each of those five.

So well done if you've got something like this.

And now we're gonna move on to the second part of the lesson and in it we're going to discuss the principle of moments using the data we collected from the investigation and then try to apply that principle in a range of situations.

So let's go on with that.

What we're going to do is analyze the situations where they're being balanced.

So we're gonna look at the combinations that cause that ruler to balance by finding the total clockwise moments and the total anticlockwise moments.

What I mean by that is I'm going to calculate the moments trying to turn it clockwise and anticlockwise separately and compare them.

So, the anticlockwise moments in this diagram would be the sum of the moments caused by each of those weights.

So I've got two weights of W and they're different distances away from the pivot that are marked there.

So my anticlockwise moments would be W times d2 added to W times d3.

My clockwise moments were a bit simpler because I've only got one distance involved there on the weight of 2W so my clockwise moments would be two times W times d1.

So let's do a few examples of that.

So here I've got a balanced ruler and we're gonna calculate the clockwise and anticlockwise moments.

So, let's start with the anticlockwise moments.

So my anticlockwise moments, I write out a statement of what I'm doing and then I sum the anticlockwise moments.

And as you can see I've got two separate masses and two different distances and I like to put each mass of distance in brackets for clarity here.

So I've got 0.

5 Newtons times 0.

1 meter because that's the distance away from the pivot.

And I add that to 0.

5 meters times 0.

3 meters and calculate that.

That gives me two separate moments, 0.

5 Newton meters and 0.

15 Newton meters and add those together to get a total of 0.

2 Newton meters anticlockwise.

The clockwise moments are a bit simpler because I've only got one mass on that side.

So I've got 0.

5 Newtons downwards and 0.

4 meters and I multiply those together.

And again, my clockwise moments of 0.

2 Newton meters.

Okay, I'd like you to try and calculate a moment.

I'd like know the total clockwise moments on this balanced ruler.

So I don't need to know anything about the anticlockwise moments, just the total clockwise moments please.

So pause the video, work that out, and restart.

Welcome back.

Well, your answer should be 0.

25 Newton meters.

If you calculate the clockwise moments, I've got two masses on that side.

We've got 0.

5 Newtons and it's 0.

30 meters away from the pivot.

And I've got another 0.

5 Newtons and that's 0.

20 meters from the pivot.

So, calculating that gives me a total of 0.

25 Newton meters.

Well done if you got that.

If you analyze all of the results from your experiment, you'll find that the ruler is balanced when the total clockwise moments are equal to the total anticlockwise moments.

So every one of your results should show that.

So I've got a situation like this and in this situation if I calculate the clockwise and the anticlockwise moments, they're both 0.

4 Newton meters.

And in this situation that ruler will be balanced.

But if I put another mass on and that changes the moment, so I've still got 0.

4 anticlockwise, but now I've got 0.

8 Newton meters clockwise, that ruler is not balanced.

So if the moments are different, the ruler is unbalanced.

If the moments are equal to each other in opposite directions, then there's no overall change.

So there's a balanced ruler.

What I'd like you to do is to use the idea to decide which of the following beams are balanced and these figures aren't drawn to scale.

So I've got different forces and different distances from the pivot and some of these are balanced and some aren't.

So, I'd like you to decide that by calculating the clockwise and anticlockwise moments and seeing if they're the same or not.

So pause the video, work that out, and restart please.

Welcome back.

Hopefully you selected beams A and C as being balanced.

So let's have a look at the moments for each of them.

And in the first one is situation A, I've got one Newton meter clockwise and one Newton meter anticlockwise.

So that one's balanced.

And in situation C, again, I've got two Newton meters clockwise and two Newton meters anticlockwise.

So that one's balanced.

But in situation B, I've got different moments in different directions.

So I've got clockwise moments of 4.

8 Newton meters and anticlockwise moments of 4.

2 and they're different so that ruler isn't balanced.

Well done if you selected A and C.

That idea leads us to the principle of moments and the principle of moments tells us when a beam is balanced.

And I've got it written out here in a box.

If an object is in equilibrium, the sum of the clockwise moments is equal to the sum of the anticlockwise moments.

So, I've got this situation and I'll mark all the distances and forces on it and that means that if I do the clockwise moments and anticlockwise moments, you can see that they're equal.

So F1d1, the force times the distance one added to the force two times distance two is equal to force three times distance three.

We mustn't forget that to be in equilibrium as well the forces in opposite directions must be balanced.

So, I've got three downward forces though, F1, F2, and F3.

So there must be an upwards force that counteracts them for it to be in equilibrium.

So, there's no resultant force acting on the beam as well and that must mean that this, those three forces plus the weight of the ruler must be balanced by a normal reaction force from the pivot.

And I've drawn that on the diagram as are there.

So, R is equal to the weight, plus the three forces of the masses on top of the ruler.

We can use the principle of moments to find unknown forces or distances.

So I've got an example here.

I've got a balanced beam, but that balanced beam has one of the pieces of information missing from it and that's the distance I've placed one of the forces from the pivot.

So what I'm going to do is to try and find that distance x.

And to do that I follow these stages.

First thing I do is I can calculate the anticlockwise moments because I've got a force and a distance in the anticlockwise direction so I can actually calculate that moment.

I write down my equation, substitute in the two values and that gives me an anticlockwise moment of 16 Newton meters.

Now I know that the beam is balanced.

So, the next stage is to use the anticlockwise moment to say that the clockwise moment must also be 16 Newton meters.

So, I can use that fact for the beam being balanced and use it to find the clockwise moment to find a distance.

So, I write up the equation again, but this time I rearrange it.

So, I'll start with M equals F times x, but I rearrange it to give x is equal to M divided by F.

So the distance is the moment divided by the force.

And if I substitute the numbers there, I quite simply calculate x as being 0.

5 meters.

Okay, now it's time for you to have a try at that.

I've got a beam here and this one's balanced.

And what I'd like you to do is find the size of the force W.

So, I've got clockwise moments and anticlockwise moments and they're equal to each other.

So I'd like you to find that missing force W please.

So pause the video, work out what that force is, and restart.

Welcome back.

Hopefully you selected 6.

4 Newtons.

And to work that out, what I did was calculate the clockwise moments using the information on the diagram and that gives me clockwise moments of 1.

60 Newton meters.

And then I know the anticlockwise moments must be equal.

So we can use the anticlockwise moments to calculate W like this, write out the equation.

The weight, W, must be equal to the moments, the anticlockwise moments, divided by that distance of 0.

25 meters.

And that gives me 6.

4 Newtons.

Well done if you got that.

You can use the principle of moments no matter how many weights and distances are involved in the balanced beam.

So I've got a slightly more complicated situation here and I'm going to show you how to calculate the missing weight, W.

So this is a balanced beam.

So to find the force W, I'm going to calculate the anticlockwise moments first because I've got all the information in the anticlockwise direction.

So, the anticlockwise moments, the moments must be equal to the forces times the distances.

So I've got 10 Newtons and that's 0.

5 meters away from that pivot.

And then I've got another five Newtons, but that's 0.

8 meters away from the pivot.

'Cause if you look careful at the diagram, it's 0.

5 meters and another 0.

3 meters.

So I've been very careful with that.

So I'm gonna calculate those anticlockwise moments now and doing the math that gives me anticlockwise moments of nine Newton meters.

Now the clockwise moments must be equal to that because they are equal and opposite for the beam to be balanced.

So the clockwise moments, I can use those to find the missing weight, W.

So, the force would be equal to the moment, divided by the distance or W equals M divided by d, substituting that value for the moments I calculated earlier and the distance of 0.

6 meters and that gives me a missing weight of 15 Newtons.

And now it's time for you to have a go at that process.

So I've got another balanced beam here, I'd like you to find the distance x.

So, remember you're calculating the moment on one side and using that to find missing values on the other.

So pause the video, work out that distance x, and then restart please.

Welcome back.

Hopefully you selected option B, 0.

4 meters.

All I needed to do to find that was calculate the clockwise moments using all the information on the clockwise side of the beam, in the right-hand side there.

And that gives me a total clockwise moments of 3.

6 Newton meters.

And I use that value for the clockwise moments to find x.

So that x, that distance x must be the moment divided by the force.

And I've got both of those, put those into the equation and I get no 0.

4 meters.

Well done if you got that.

And now it's time for the final task of the lesson and what I'd like you to do is this.

I'd like you to state two conditions required for an object to be in equilibrium and then I'd like you to look at the two diagrams below and find the missing values.

So for the left hand diagram it's find the distance x.

And for the right hand diagram, I'd like you to find forces W and R, the normal reaction force at the pivot there.

So pause the video, work out your answers to that, and restart please.

Welcome back.

Well, here's the two conditions required for an object to be in equilibrium.

The sum of the clockwise moments must be equal to the sum of the anticlockwise moments and there must be no resultant force acting on the object.

Well done if you got both of those.

And here's the answer to the first of those scenarios about the beams in equilibrium.

We're finding the missing values here and finding the distance x.

So you find the anticlockwise moments and use those to find the clockwise moments.

And that gives me a distance of 0.

5 meters.

Well done if you got that one.

And here's the second of those.

And again, we find the clockwise moments and that gives me a clockwise moments of 5.

2 Newton meters and then the anticlockwise moments.

So that gives me a weight W of 30 Newtons.

But we also had to find the reaction force, R, that's the upwards force.

Well, if you add all the down forces together, that gives a total downward force of 32 Newton, sorry, 32 Newtons.

So there must be an upwards force of 32 Newtons and that's a reaction force.

Well done if you got both of those.

And we've reached the end of the lesson now, and here's a summary of all the information we've learned.

We've got the definition of a moment.

It's the force times the distance, normal to the direction of the force.

And we found the principle of moments and that states that for an object in equilibrium, the sum of the clockwise moments is equal to sum of the anticlockwise moments.

Any unbalanced moments will cause an object to start to rotate and it won't be in equilibrium.

It'll rotate in the direction of the largest total moment.

Well done for reaching the end of the lesson.

I'll see you in the next one.