0:00:01.820,0:00:06.900
Let's consider the formula[br]for the period of a simple

0:00:06.900,0:00:11.472
pendulum of length L is[br]formula is T period.

0:00:12.560,0:00:14.080
Is equal to 2π?

0:00:15.180,0:00:21.072
Times the square root of L[br]over G&L is the length of

0:00:21.072,0:00:22.054
the pendulum.

0:00:23.220,0:00:27.114
Now on Earth, we tend to[br]regard G's being fixed it.

0:00:27.114,0:00:30.654
There is a little with[br]altitude, but we tend to

0:00:30.654,0:00:32.778
think of it as being fixed.

0:00:33.810,0:00:39.998
Just supposing we had a pendulum[br]of a fixed length L, and we took

0:00:39.998,0:00:43.976
it somewhere else. Let's say the[br]moon or Mars.

0:00:44.670,0:00:45.970
Somewhere out of the solar

0:00:45.970,0:00:50.810
system. Gravity there would not[br]be the same and we might want to

0:00:50.810,0:00:54.530
measure gravity, so one of the[br]ways will be to take our

0:00:54.530,0:00:55.770
pendulum, set it swinging.

0:00:56.780,0:00:58.100
And measure the time.

0:00:59.060,0:01:03.536
Once we measured the time, we[br]could then use that to Calculate

0:01:03.536,0:01:09.504
G, but before we could do it, we[br]would need to know what G was in

0:01:09.504,0:01:14.726
terms of the rest of the symbols[br]in this formula. So we need to

0:01:14.726,0:01:17.710
rearrange this formula so that[br]it said G.

0:01:19.230,0:01:20.210
Equals.

0:01:22.200,0:01:23.559
Now it's what?

0:01:24.270,0:01:28.518
Goes instead of this question[br]mark that we're going to have a

0:01:28.518,0:01:33.120
look at in this video. We're[br]going to be looking at how we

0:01:33.120,0:01:36.660
transform formally, how we move[br]from an expression like this.

0:01:37.700,0:01:39.218
To another expression.

0:01:40.080,0:01:44.780
Involving exactly the same[br]variables, but one that helps us

0:01:44.780,0:01:49.950
answer the questions that were[br]after in terms of a different

0:01:49.950,0:01:52.508
variable. To do this?

0:01:53.410,0:01:57.840
Transformation of formula will[br]need all the techniques that we

0:01:57.840,0:02:02.270
had in terms of solving[br]equations, so the video solving

0:02:02.270,0:02:07.586
linear equations in one variable[br]might be very useful to have a

0:02:07.586,0:02:12.016
look at, but let's just recap by[br]having a look.

0:02:13.570,0:02:15.990
A simple linear equation.

0:02:16.710,0:02:21.336
So the one going to take[br]three X +5.

0:02:22.860,0:02:28.410
Equals 6[br]- 3 * 5

0:02:28.410,0:02:31.185
- 2 X.

0:02:32.740,0:02:37.206
Now we've got this equation we[br]want to solve it for X, so our

0:02:37.206,0:02:39.758
aim is to end up with X equals.

0:02:40.910,0:02:45.206
So one of the first things[br]we would do is multiply out

0:02:45.206,0:02:45.922
the bracket.

0:02:48.110,0:02:53.258
So we have minus three times by[br]5 is minus 15.

0:02:54.130,0:03:00.439
Minus three times by minus[br]2X is plus 6X.

0:03:01.650,0:03:09.030
Next, we can simplify this[br]bit. Here three X +5

0:03:09.030,0:03:16.410
equals 6X, and now six[br]takeaway 15 is minus 9.

0:03:17.760,0:03:24.368
Now, one of the things we would[br]want to do is to try and get all

0:03:24.368,0:03:30.150
the ex is together. So we were[br]three X here under 6X there, so

0:03:30.150,0:03:34.693
will take 3X away from each[br]side, so that's three X

0:03:34.693,0:03:38.823
takeaway. Three X +5 equals 6X[br]takeaway, 3X minus 9.

0:03:39.630,0:03:46.013
3X takeaway 3X the X is go.[br]We've known left and then we've

0:03:46.013,0:03:52.396
got this. Five is equal to six X[br]takeaway 3X. That's three X

0:03:52.396,0:03:53.869
again takeaway 9.

0:03:55.400,0:04:00.524
Now we need to get the constant[br]terms, the numbers together. So

0:04:00.524,0:04:07.783
to do that we would add 9 to[br]each side. So we 5 + 9 is equal

0:04:07.783,0:04:10.772
to three X minus 9 + 9.

0:04:11.570,0:04:17.078
So again, we're doing to the[br]same thing to both sides. Here

0:04:17.078,0:04:23.045
we took three X away from each[br]side. Here we're adding nine to

0:04:23.045,0:04:29.930
each side, so five and nine is[br]14 equals 3X, and we minus 9 +

0:04:29.930,0:04:34.520
9.0. Now we need to divide both[br]sides by three.

0:04:35.340,0:04:41.724
So we have 3X over three an 14[br]over 3, dividing both sides by

0:04:41.724,0:04:46.740
three, and so those three[br]canceled and we're left with X

0:04:46.740,0:04:51.756
equals 14 over 3. Now I've[br]written this out very fully.

0:04:51.756,0:04:56.772
I've written out every step that[br]I've said, but we wouldn't

0:04:56.772,0:05:00.876
normally expect to see all of[br]that written down.

0:05:02.100,0:05:03.720
Starting from here.

0:05:07.750,0:05:11.974
The next line we'd expect[br]to see is this one because

0:05:11.974,0:05:16.198
we say Take 3X away from[br]both sides and we'd expect

0:05:16.198,0:05:18.502
to see that as the result.

0:05:19.540,0:05:24.714
Then we'd say add 9 to both[br]sides, and So what we would

0:05:24.714,0:05:28.296
expect to see having done that[br]would be that.

0:05:28.930,0:05:31.990
And then we, say, divide both[br]sides by three.

0:05:34.030,0:05:38.398
And so we see that So what we[br]would see written down in our

0:05:38.398,0:05:41.830
exercise book or on a piece of[br]paper that carried the

0:05:41.830,0:05:44.638
solution to this equation will[br]be something like this.

0:05:45.830,0:05:49.810
OK, we've been through the[br]steps now of solving an

0:05:49.810,0:05:52.994
equation. These techniques,[br]particularly idea of a balance

0:05:52.994,0:05:57.770
of keeping the same on both[br]sides by doing the same thing

0:05:57.770,0:06:02.944
to both sides is what we're[br]going to do when we look at

0:06:02.944,0:06:04.536
the transformation of[br]formally.

0:06:07.060,0:06:14.460
So let's begin with our[br]first formula V equals U

0:06:14.460,0:06:20.155
plus AT. And the variable[br]that we're going to try and

0:06:20.155,0:06:21.310
find is TI.

0:06:22.320,0:06:28.464
T is the variable we're going to[br]find what is T expressed?

0:06:29.130,0:06:33.486
In terms of the rest of the[br]letters or symbols that there

0:06:33.486,0:06:37.842
are in this formula well, in[br]order to model this, what I'm

0:06:37.842,0:06:42.198
going to do is I'm going to[br]write down an equation which

0:06:42.198,0:06:44.739
looks like this, but it just got

0:06:44.739,0:06:51.595
numbers in. With a T there as[br]the unknown. So what we might

0:06:51.595,0:06:56.445
have is something like this.[br]Seven for V5 for you.

0:06:57.040,0:07:02.026
Let's say a tool[br]for A and then T.

0:07:03.440,0:07:08.432
Now what would we do to solve[br]this equation? The first thing

0:07:08.432,0:07:15.088
we try and do is get the TS on[br]their own and so to do that,

0:07:15.088,0:07:20.912
we take 5 away from each side.[br]So this would be 2. Over here

0:07:20.912,0:07:25.904
equals 2 T, so having done[br]that there, let's do it here.

0:07:25.904,0:07:29.648
V minus U taking you away from[br]each side.

0:07:31.280,0:07:36.190
Now we would divide both sides[br]by two over here.

0:07:39.400,0:07:46.600
In order to end up with just T[br]on its own, so let's do the

0:07:46.600,0:07:52.840
same. Here. A is what multiplies[br]by T, so we need to divide

0:07:52.840,0:07:59.080
everything on this side V minus[br]U over a equals T. That's it.

0:07:59.080,0:08:05.320
Notice everything here is over[br]a, not just one part of it, but

0:08:05.320,0:08:07.240
both the whole expression.

0:08:07.470,0:08:10.070
The minus U over a.

0:08:11.800,0:08:16.665
Let's take another[br]one. V squared equals

0:08:16.665,0:08:20.140
U squared plus 2A S.

0:08:21.190,0:08:24.778
And this time, let's say the[br]symbol that I'm going to try and

0:08:24.778,0:08:28.366
find all the variables that I'm[br]going to try and find in terms

0:08:28.366,0:08:30.022
of all the others, is you.

0:08:31.990,0:08:36.577
So that's the case. Again,[br]I'm going to write down an

0:08:36.577,0:08:39.913
equation over here which[br]looks like this one.

0:08:41.090,0:08:42.209
So let's have.

0:08:43.450,0:08:44.620
25

0:08:46.060,0:08:48.370
equals U squared.

0:08:49.360,0:08:51.660
+9.

0:08:53.180,0:08:58.262
Weather 9 is in place of this[br]lump here of algebra.

0:08:58.850,0:09:04.752
So what would we do here? Our[br]first step would be to take

0:09:04.752,0:09:10.654
nine away from both sides. So[br]let's do that. 25 - 9 equals

0:09:10.654,0:09:16.556
U squared and that gets us EU[br]squared on its own. So let's

0:09:16.556,0:09:21.550
do that over here. Let's take[br]this lump away from both

0:09:21.550,0:09:26.544
sides. 20 squared minus 2A S[br]is equal to U squared.

0:09:28.150,0:09:33.535
Now what I would want to do now[br]is to take the square root of

0:09:33.535,0:09:37.843
both sides 'cause I want just[br]you and here I've got you

0:09:37.843,0:09:43.228
squared. So I need the square[br]root of 25 - 9 and I want the

0:09:43.228,0:09:47.895
square root of all of that not[br]square root of 25 minus the

0:09:47.895,0:09:53.280
square root of 9. I want the[br]square root of 25 - 9, so again

0:09:53.280,0:09:58.306
I've got to do the same over[br]here I want the square root of.

0:09:58.430,0:10:00.620
All. Love it.

0:10:05.400,0:10:10.847
Let's just check why I need the[br]square root of all of this.

0:10:12.030,0:10:18.675
25 - 9 is 16, so that 16 equals[br]U squared. Take the square root

0:10:18.675,0:10:24.434
of both sides. Four is equal to[br]you and we're happy. That's the

0:10:24.434,0:10:30.566
answer. But what if I do the[br]square root of 25 minus the

0:10:30.566,0:10:36.488
square root of nine? Well,[br]that's 5 - 3 is 2, which is not

0:10:36.488,0:10:41.564
the answer very definitely not[br]the answer, so we need to take

0:10:41.564,0:10:46.217
the square root of everything,[br]not just each little piece. And

0:10:46.217,0:10:52.139
so this over here has to be the[br]square root of all of that.

0:10:55.720,0:11:01.456
I've been working so far with[br]the formula that are for uniform

0:11:01.456,0:11:05.280
acceleration, so let's continue[br]with that and take.

0:11:05.890,0:11:13.534
Another one of them S equals[br]UT plus 1/2 AT squared, and

0:11:13.534,0:11:16.082
this time it's a.

0:11:17.630,0:11:21.914
Then I'm going to be looking for[br]can I rearrange this formula?

0:11:21.914,0:11:24.770
Can I transform it so it says a

0:11:24.770,0:11:32.063
equals? Again, let me model it[br]with an equation. Let's say that

0:11:32.063,0:11:39.945
S is 21, but U times by[br]T is 15 + 1/2 of a

0:11:39.945,0:11:41.634
Times by 9.

0:11:42.870,0:11:49.240
And it's this a the time after[br]first of all. Let's isolate the

0:11:49.240,0:11:55.610
term with the unknown in it. So[br]that's the ater. So let's take

0:11:55.610,0:12:02.960
15 away from both sides. So I'm[br]21 - 15 is equal to 1/2 of

0:12:02.960,0:12:09.820
a times by 9. So again, let's do[br]that here. Let's take this lump

0:12:09.820,0:12:12.760
away. S minus Utah equals 1/2.

0:12:12.870,0:12:18.525
AT squared, so again, my first[br]steps are to try and isolate the

0:12:18.525,0:12:20.700
variable. The term that I'm

0:12:20.700,0:12:25.922
looking for. Now this one[br]looks a bit complicated. We

0:12:25.922,0:12:31.434
got a half. There would be[br]nice to get rid of the half,

0:12:31.434,0:12:36.946
so let's multiply both sides[br]by two. If we do it at this

0:12:36.946,0:12:43.306
side, that's just a times by[br]9. We do it this side 2 * 21

0:12:43.306,0:12:48.818
- 15 and it multiplies all[br]of it. So let's do the same

0:12:48.818,0:12:53.482
here. Multiply both sides by[br]2, two times S minus UT.

0:12:54.780,0:12:58.338
Equals AT squared.

0:13:00.000,0:13:06.104
Going back to the equation we[br]got equals a Times by 9. I just

0:13:06.104,0:13:12.208
want a on its own and so I must[br]divide both sides by 9.

0:13:15.470,0:13:18.935
Here the thing that's doing[br]the multiplying it's T

0:13:18.935,0:13:23.555
squared. So let's divide both[br]sides by T squared and in the

0:13:23.555,0:13:27.790
same way as I divided[br]everything by 9, I've got to

0:13:27.790,0:13:30.100
divide everything here by T[br]squared.

0:13:37.740,0:13:40.400
And that gives me a.

0:13:42.540,0:13:47.300
Notice I haven't made any[br]effort to cancel because T

0:13:47.300,0:13:52.536
is not a common factor. It[br]is not a common factor.

0:13:54.770,0:14:00.590
No, we've been working with the[br]equations that are to do with

0:14:00.590,0:14:05.042
uniform acceleration. But there[br]are some other kinds of

0:14:05.042,0:14:09.434
equations that we need to gain[br]practice at and in order to

0:14:09.434,0:14:13.460
develop these skills, I'm going[br]to start with some made up

0:14:13.460,0:14:16.388
equations that don't have[br]physical applications in the

0:14:16.388,0:14:20.414
real world, but we will come[br]back once we've developed those

0:14:20.414,0:14:24.074
skills to looking at some real[br]equations that do contain

0:14:24.074,0:14:27.734
physical applications that we[br]will be able to transform, but

0:14:27.734,0:14:32.492
we need to develop some skills[br]to begin with. So first of all.

0:14:32.670,0:14:38.440
Let's take this expression,[br]let's call it rather than a

0:14:38.440,0:14:39.017
formula.

0:14:40.840,0:14:46.990
So then we have Y times 2X plus[br]one equals X Plus One and the

0:14:46.990,0:14:52.320
thing that we're going to be[br]after is X. Can we rearrange it

0:14:52.320,0:14:54.370
so it says X equals?

0:14:55.030,0:14:58.998
Well gain, let me[br]try and model this.

0:15:00.600,0:15:05.495
With an equation. So all[br]I need to do to model

0:15:05.495,0:15:09.945
this as an equation is[br]replaced the Y by three.

0:15:11.470,0:15:16.030
And if I was to solve this[br]inequation, my first step would

0:15:16.030,0:15:20.970
be to multiply out the bracket.[br]So let's do that. 6X plus three

0:15:20.970,0:15:23.250
is equal to X plus one.

0:15:23.970,0:15:30.273
So let's do that over here.[br]Multiply out this bracket. So

0:15:30.273,0:15:33.138
that's two XY Plus Y.

0:15:33.850,0:15:37.216
Is equal to X plus one.

0:15:38.730,0:15:44.749
Our next step with the equation[br]will be to get all the excess

0:15:44.749,0:15:50.768
together. So I would take X away[br]from both sides, so that would

0:15:50.768,0:15:57.250
be 6X takeaway X +3, and taking[br]the X away from this side just

0:15:57.250,0:16:03.269
leaves me with one. So let's do[br]that here. Let's take this X

0:16:03.269,0:16:08.825
away from both sides, so have[br]two XY minus X Plus Y.

0:16:09.060,0:16:10.648
Is equal to 1.

0:16:12.030,0:16:18.390
Now I would naturally want to[br]combine these two in some way.

0:16:18.390,0:16:21.570
6X minus X is just 5X.

0:16:22.790,0:16:28.880
I can't really do that at this[br]side, but what I can do is

0:16:28.880,0:16:34.535
gather together the terms in X[br]by taking out X as a common

0:16:34.535,0:16:40.190
factor. So we take out X from[br]this as a common factor. The

0:16:40.190,0:16:42.800
other factor is 2 Y minus.

0:16:44.010,0:16:46.946
And taking X out[br]of there is one.

0:16:48.330,0:16:52.490
Plus Y equals[br]1.

0:16:54.800,0:16:59.935
What now? Well now I've got my[br]ex is together over here. Let's

0:16:59.935,0:17:05.860
write this as 5X plus 3 equals[br]1. My next step will be to leave

0:17:05.860,0:17:10.995
the X term on its own by taking[br]three away from each side.

0:17:13.190,0:17:19.538
So let's do that here. Lead[br]the Exterm on its own by

0:17:19.538,0:17:24.828
taking the other term. That's[br]why away from both sides.

0:17:30.200,0:17:34.796
Now I'm just going to simplify[br]this. Five X equals minus two,

0:17:34.796,0:17:40.541
and in order to solve this to[br]get a value of XI need to divide

0:17:40.541,0:17:45.520
both sides by this number 5. So[br]that's X equals minus two over

0:17:45.520,0:17:51.265
5. So if I come back to this[br]this term in the bracket, two Y

0:17:51.265,0:17:56.244
minus one is the term that's[br]multiplying the X, and so I need

0:17:56.244,0:18:03.498
to divide. Both sides by so X[br]is equal to 1 minus Y over

0:18:03.498,0:18:09.372
2Y plus. Sorry made a mistake[br]there. Two Y minus one.

0:18:12.020,0:18:16.676
I just look again at what we've[br]done here. We've mimic the

0:18:16.676,0:18:20.556
solving of an equation. First,[br]we multiplied out the brackets.

0:18:22.300,0:18:25.606
Then we got the terms together

0:18:25.606,0:18:29.198
in X. The variable[br]that we were after.

0:18:31.310,0:18:33.760
Having got those terms together.

0:18:34.540,0:18:38.566
We then isolated them so that[br]they were on their own.

0:18:39.970,0:18:43.870
So we need to bear that in mind[br]and follow it through.

0:18:47.190,0:18:54.165
Let's take another expression Y[br]over Y plus X.

0:18:55.770,0:18:58.710
+5 is equal to X.

0:18:59.440,0:19:04.480
This time. We're going to be[br]finding why we're going to be

0:19:04.480,0:19:08.562
getting Y equals, and we want[br]some lump of numbers and X is

0:19:08.562,0:19:09.504
over this sigh.

0:19:11.180,0:19:19.064
So let's model this with an[br]equation Y over Y plus 3

0:19:19.064,0:19:23.006
+ 5 is equal to 3.

0:19:24.580,0:19:28.100
Look at this side. Here we've[br]got an algebraic fraction.

0:19:28.950,0:19:33.617
Y over Y plus 3Y plus three is[br]the denominator. It's in the

0:19:33.617,0:19:35.053
bottom of the fraction.

0:19:35.840,0:19:41.885
So in order to get rid of that,[br]we need to multiply everything

0:19:41.885,0:19:46.535
in this equation by this[br]denominator. So I'm going to

0:19:46.535,0:19:53.045
write that out in full Y over Y,[br]plus three. Put that in a

0:19:53.045,0:19:55.370
bracket times Y plus 3.

0:19:56.680,0:19:59.888
Plus five times Y.

0:20:00.290,0:20:06.530
Three is equal to three times Y[br]plus three. Let me just

0:20:06.530,0:20:13.290
emphasize we have to do the same[br]thing to both sides, so here

0:20:13.290,0:20:19.530
I've had to multiply everything[br]on both sides of the equation by

0:20:19.530,0:20:22.130
this term Y plus 3.

0:20:23.230,0:20:27.190
Here those will cancel out,[br]just leaving me with why, so

0:20:27.190,0:20:30.790
let's do that at this site.[br]Let's multiply everything by

0:20:30.790,0:20:35.470
this term Y plus X so we know[br]this first one. When we've

0:20:35.470,0:20:39.790
multiplied it by wiper sex is[br]just going to give us Why.

0:20:41.390,0:20:48.090
Plus five times by Y[br]plus X is equal to

0:20:48.090,0:20:52.110
X times by Y plus[br]X.

0:20:54.000,0:20:57.344
Coming back to the equation[br]here, faced with a lot of

0:20:57.344,0:21:00.688
brackets, we know the first[br]thing we would do is multiply

0:21:00.688,0:21:03.120
out the brackets. So let's do[br]that, why?

0:21:04.350,0:21:11.938
Plus five times by Y is 5[br]Y five times by three is 15

0:21:11.938,0:21:19.526
is equal 2 three times by Y[br]is 3 Y and three times by

0:21:19.526,0:21:21.152
three is 9.

0:21:23.210,0:21:25.670
Do the same here, why?

0:21:27.050,0:21:34.699
+5 Y.[br]Five times by X Plus 5X is

0:21:34.699,0:21:41.940
equal to X times by Y is[br]XY&X times by X is X.

0:21:42.600,0:21:43.230
Square.

0:21:45.390,0:21:49.688
On this side now we would like[br]to get all of our wise together.

0:21:50.680,0:21:53.944
So we've Y plus

0:21:53.944,0:22:00.138
5Y6Y. Take away 3 Y[br]so we've six. Why

0:22:00.138,0:22:05.367
already and we're going[br]to take away 3Y. Plus

0:22:05.367,0:22:11.177
15 is equal to 9, so[br]let's gather all the

0:22:11.177,0:22:15.825
wise together Y +5. Y[br]is 6 Y.

0:22:17.340,0:22:23.332
I need to bring this term over,[br]so I'll take XY away from both

0:22:23.332,0:22:24.616
sides minus XY.

0:22:25.410,0:22:29.100
And then let's just write down[br]the other terms.

0:22:30.940,0:22:37.002
Now here I simplify this. I'd[br]have six Y takeaway 3 Y and that

0:22:37.002,0:22:43.064
would just leave me with three Y[br]plus 15 equals 9. I can't do

0:22:43.064,0:22:49.126
that here, but the thing that I[br]can do is bring them much closer

0:22:49.126,0:22:51.291
together by taking out why.

0:22:52.290,0:22:58.192
The thing that I'm after as a[br]common factor. So let's take why

0:22:58.192,0:23:04.094
out of these two terms. Why[br]brackets? Why times by 6 Y means

0:23:04.094,0:23:11.358
I must have a 6 in their minus[br]XY means I must have a minus X

0:23:11.358,0:23:14.082
there plus 5X equals X squared.

0:23:15.380,0:23:22.335
Now over here I'd isolate this[br]term in why by taking 15 away

0:23:22.335,0:23:29.825
from both sides 3 Y is equal[br]to. Now this is 9 - 15,

0:23:29.825,0:23:36.780
so I've got to do the same[br]here. Take 5X away from each

0:23:36.780,0:23:43.735
side, isolating this term in YY[br]times 6 minus X is equal to

0:23:43.735,0:23:45.340
X squared minus.

0:23:45.480,0:23:47.170
5X.

0:23:48.930,0:23:54.824
This side now I've got 3 Yi,[br]just need Y so I would divide

0:23:54.824,0:23:59.876
everything on this side by[br]three. So I've got to do the

0:23:59.876,0:24:04.507
same with this. I've got to[br]divide everything on this side

0:24:04.507,0:24:09.980
by this term 6 minus X so we[br]have X squared minus 5X.

0:24:10.640,0:24:14.620
Divided by 6 minus X.

0:24:15.960,0:24:21.144
Put it all in a bracket to keep[br]it together. Notice I'm

0:24:21.144,0:24:25.464
attempting no canceling 'cause[br]there is no common factor in

0:24:25.464,0:24:29.784
this numerator and this[br]denominator. They do not share a

0:24:29.784,0:24:34.536
common factor, so that's my[br]answer and I finished. I've got

0:24:34.536,0:24:39.288
Y equals a lump of algebra[br]involving X is and numbers.

0:24:42.980,0:24:48.656
We started with a real formula.[br]We started with the formula for

0:24:48.656,0:24:50.075
a simple pendulum.

0:24:50.880,0:24:57.075
Of length Lt equals[br]2π square root.

0:24:57.630,0:25:04.270
L over G and the problem that we[br]posed was what was G in terms of

0:25:04.270,0:25:06.900
T? So G.

0:25:07.990,0:25:10.909
After G equals.

0:25:13.230,0:25:17.778
Let's have a look at a simple[br]equation. Let's say it's 10

0:25:17.778,0:25:22.705
equals 2π and I'll keep the two[br]Pike's. Part is just a number,

0:25:22.705,0:25:24.979
so 2π is just a number.

0:25:26.360,0:25:30.950
Square root[br]of 3 over G.

0:25:32.200,0:25:37.030
Now as an equation, this one is[br]a bit tricky because the G is

0:25:37.030,0:25:41.170
trapped inside this square root[br]sign. We need to bring it out

0:25:41.170,0:25:46.000
and the way to get a square root[br]sign to disappear, so to speak,

0:25:46.000,0:25:50.140
is to square both sides, reverse[br]the operation if you like, so

0:25:50.140,0:25:56.573
will square. Both sides of this[br]equation so 10 multiplied by

0:25:56.573,0:26:02.897
itself. I'm going to write it as[br]10 squared equals 2π multiplied

0:26:02.897,0:26:09.221
by itself, 'cause we're squaring[br]the whole of both sides times by

0:26:09.221,0:26:11.329
a nice square this.

0:26:11.920,0:26:14.830
Three over G.

0:26:16.450,0:26:21.546
Now we need to do the same here,[br]square the whole of both sides.

0:26:22.420,0:26:25.306
So that will be T squared.

0:26:26.890,0:26:33.208
2π all squared[br]L over G.

0:26:35.040,0:26:38.648
Coming back to the equation,[br]jeez, in the denominator I don't

0:26:38.648,0:26:43.568
want it there. I want it to say[br]G equals I want G upstairs, so

0:26:43.568,0:26:48.488
to speak, so I have to get rid[br]of it out of the denominator and

0:26:48.488,0:26:52.096
so to do that I must multiply[br]both sides by G.

0:26:52.950,0:27:00.066
So I've got 10 squared times.[br]G is equal to 2π squared

0:27:00.066,0:27:07.182
times three, so let's do that[br]here. Multiply both sides by G.

0:27:07.182,0:27:14.298
So I have T squared times[br]by G is equal to 2π

0:27:14.298,0:27:17.263
all squared times by L.

0:27:19.500,0:27:26.700
Gee, I want on its own. I want[br]to say G equal, so I must divide

0:27:26.700,0:27:33.000
by this thing 10 squared. So G[br]is equal to 2π all squared times

0:27:33.000,0:27:35.250
3, all over 10 squared.

0:27:36.320,0:27:43.376
And I can work that out. Let's[br]do the same with this G equals

0:27:43.376,0:27:49.928
2π all squared times by L, and[br]I'm getting rid of this T

0:27:49.928,0:27:53.456
squared by dividing everything[br]by T squared.

0:27:55.800,0:27:58.544
And we can leave it like that.

0:27:59.350,0:28:05.426
We can make it look a little[br]bit nicer if we want to buy

0:28:05.426,0:28:09.332
spotting that we've got a[br]square here under Square

0:28:09.332,0:28:13.672
there, and it might be nice[br]to combine those two.

0:28:13.672,0:28:17.578
Squaring process is by[br]writing 2π over T all

0:28:17.578,0:28:19.314
squared times by L.

0:28:20.410,0:28:23.785
Basically this is the answer[br]that we finished with.

0:28:24.700,0:28:27.890
Let's take another couple of.

0:28:29.280,0:28:33.804
Examples of real formula that we[br]might want to be able to

0:28:33.804,0:28:38.705
manipulate. So let's have a look[br]at the lens Formula One over F.

0:28:39.540,0:28:41.460
Is equal to one over U.

0:28:43.030,0:28:44.098
This one over V.

0:28:44.930,0:28:46.578
Let's say it's you.

0:28:48.120,0:28:50.706
That we're after we want you

0:28:50.706,0:28:55.655
equals. Well, let me write[br]down an equation.

0:28:58.550,0:28:59.260
Similar.

0:29:00.720,0:29:05.164
One over 3 equals 1 over U plus[br]one over 5.

0:29:06.650,0:29:12.161
I want to isolate this term[br]first in one over you.

0:29:13.380,0:29:18.983
To do that, I'm going to take[br]this away from both sides. 1/3

0:29:18.983,0:29:21.569
takeaway 150 is equal to one

0:29:21.569,0:29:26.862
over you. So let's do that.[br]Isolate this term in you.

0:29:29.000,0:29:33.680
And so to do that, we take away[br]one over V from both sides, so

0:29:33.680,0:29:34.928
we want over F.

0:29:35.530,0:29:39.523
Take away one over V is equal to[br]one over you.

0:29:40.930,0:29:45.493
Now, faced with problems[br]like this, it's very, very

0:29:45.493,0:29:49.042
tempting to simply turn[br]everything upside down.

0:29:50.290,0:29:56.800
Well, I just have a think about[br]that. This says 1/3 - 1/5. Now a

0:29:56.800,0:29:58.970
third is bigger than 1/5.

0:30:00.810,0:30:05.220
So 1/3 - 1/5 is a positive[br]number, so you at the very least

0:30:05.220,0:30:06.795
has got to be positive.

0:30:07.860,0:30:12.080
Watch what happens if I just[br]turn everything upside down.

0:30:12.710,0:30:19.670
3 - 5 equals U, which tells me[br]that minus two is equal to you.

0:30:19.670,0:30:25.702
But we just agreed that you had[br]to be positive and not negative.

0:30:25.702,0:30:27.558
You can't do that.

0:30:28.380,0:30:32.020
These are fractions, and so[br]because their fractions we have

0:30:32.020,0:30:36.388
to combine them as fractions,[br]which means we have to find a

0:30:36.388,0:30:40.028
common denominator and a common[br]denominator for three and five

0:30:40.028,0:30:44.032
is a number that both three and[br]five will divide into.

0:30:44.910,0:30:48.627
Easiest number is 3 times by 5.

0:30:49.270,0:30:55.500
So three Zing to three times by[br]5, which is 15 goes five times.

0:30:55.500,0:31:02.175
So I found multiplied 3 by 5. I[br]must multiply by this one by 5,

0:31:02.175,0:31:03.955
so that's 5 minus.

0:31:04.600,0:31:10.168
I've multiplied 5 by three, so I[br]multiply this one by three

0:31:10.168,0:31:17.128
equals one over. You know if I[br]did that here, I've got to do it

0:31:17.128,0:31:21.746
over here. So I want[br]to common denominator.

0:31:23.300,0:31:27.944
Here the common denominator I[br]talk was three times by 5, so

0:31:27.944,0:31:30.266
let's take F times by vis.

0:31:32.720,0:31:34.688
I've multiplied F.

0:31:36.230,0:31:38.960
By the so I must multiply the

0:31:38.960,0:31:40.730
one. By faith.

0:31:43.890,0:31:44.730
Minus.

0:31:45.830,0:31:49.436
I've multiplied the V by F.

0:31:50.180,0:31:53.436
So I must multiply the one by F.

0:31:58.700,0:32:04.212
Now if I come back over here,[br]let me just simplify this. This

0:32:04.212,0:32:10.148
is 2 over 15 is one over you,[br]and now I've got a complete

0:32:10.148,0:32:15.236
fraction on both sides. I can[br]turn it upside down and say

0:32:15.236,0:32:16.932
that's what you is.

0:32:18.370,0:32:23.430
Now there was a lot of[br]calculation went on here. I

0:32:23.430,0:32:28.490
can't do that calculation here,[br]but I have got a complete

0:32:28.490,0:32:31.710
fraction here so I can turn it.

0:32:34.740,0:32:35.820
Upside down.

0:32:36.850,0:32:38.198
To give me you.

0:32:39.670,0:32:44.930
So let's take as our final[br]example the time dilation

0:32:44.930,0:32:46.508
formula from relativity.

0:32:47.480,0:32:52.709
The formula is T[br]equals T, not.

0:32:53.990,0:32:55.440
Divided by.

0:32:56.630,0:33:04.206
1. Minus. V[br]squared over C squared to the

0:33:04.206,0:33:07.296
power. 1/2 or square root.

0:33:08.880,0:33:13.534
So let's take some numbers and[br]in fact what we're going to be

0:33:13.534,0:33:17.114
after is we're going to be[br]looking for this expression

0:33:17.114,0:33:21.410
here, V over C. So I just[br]write that down there. That's

0:33:21.410,0:33:26.422
what we're going to be looking[br]for. Can we get V over C in

0:33:26.422,0:33:27.854
terms of T&T Nord?

0:33:29.160,0:33:34.552
So let's have 6[br]equals 5 over 1

0:33:34.552,0:33:36.574
minus X squared.

0:33:39.170,0:33:42.798
To the half. Now solving this.

0:33:43.410,0:33:48.282
What would we do? Well, we've[br]got a square root. Let's Square

0:33:48.282,0:33:53.560
both sides in order to get rid[br]of that square root. So that

0:33:53.560,0:33:58.432
would be 6 squared equals 5[br]squared all over 1 minus X

0:33:58.432,0:34:03.666
squared. So let's do that here.[br]Square both sides.

0:34:04.320,0:34:11.074
T squared. Equals TN[br]squared all over 1

0:34:11.074,0:34:15.898
minus V squared over[br]C squared.

0:34:17.590,0:34:24.394
Now, well here the term I want[br]in X is in the denominator. I

0:34:24.394,0:34:29.254
need it upstairs, so let's[br]multiply both sides of this

0:34:29.254,0:34:34.114
equation by one minus X[br]squared. That would be 6

0:34:34.114,0:34:39.460
squared times 1 minus X[br]squared is equal to 5 squared.

0:34:40.740,0:34:42.750
So let's do that here.

0:34:43.990,0:34:49.438
T squared times, 1[br]minus V squared over

0:34:49.438,0:34:54.886
C squared is equal[br]to T Nord squared.

0:34:57.150,0:35:00.566
Now I want the X squared bit.

0:35:01.720,0:35:06.652
And it would be nice perhaps to[br]multiply out the bracket, but

0:35:06.652,0:35:11.584
there's a slightly quicker way.[br]I can divide both sides by 6

0:35:11.584,0:35:15.694
squared, and that's nice. 'cause[br]it keeps the square bits

0:35:15.694,0:35:21.448
together, so to speak. So let me[br]do that one minus X squared is

0:35:21.448,0:35:26.791
equal to 5 squared over 6[br]squared. So I'm going to do that

0:35:26.791,0:35:31.723
here. Divide both sides by T[br]squared, 1 minus V squared over

0:35:31.723,0:35:38.808
C squared. Is equal to[br]TN squared over T squared?

0:35:40.740,0:35:41.190
Now.

0:35:42.430,0:35:45.430
Running out of paper here.[br]So what I'm going to do is

0:35:45.430,0:35:48.680
turn over and write this one[br]down at the top of the next

0:35:48.680,0:35:50.180
page, and this one as well.

0:35:52.980,0:36:00.042
So we've got 1 minus V[br]squared over. C squared is

0:36:00.042,0:36:05.178
equal to T not squared over[br]T squared.

0:36:07.340,0:36:13.676
And here, with one minus X[br]squared is equal to 5 squared

0:36:13.676,0:36:15.260
over 6 squared.

0:36:17.050,0:36:23.836
It's The X squared that we're[br]after so I can add X squared

0:36:23.836,0:36:30.100
both sides, so this is one[br]equals 5 squared over 6 squared

0:36:30.100,0:36:37.408
plus X squared. So let me add[br]V squared over C squared to each

0:36:37.408,0:36:44.194
side. One is equal to T not[br]squared over T squared plus B

0:36:44.194,0:36:46.282
squared over C squared.

0:36:47.840,0:36:50.207
Now I want the X[br]squared on it so.

0:36:51.440,0:36:58.556
So take this away from both[br]sides. 1 - 5 squared over

0:36:58.556,0:37:05.672
6 squared is equal to X[br]squared, so 1 minus T, not

0:37:05.672,0:37:12.788
squared over T squared is equal[br]to V squared over C squared.

0:37:12.788,0:37:15.753
Taking this away from both

0:37:15.753,0:37:23.507
sides. Almost there now it's[br]X that I want not X squared

0:37:23.507,0:37:26.909
selects. Take the square root of

0:37:26.909,0:37:33.923
both sides. So do the[br]same here. The oversee that I

0:37:33.923,0:37:39.593
want, so let's take the square[br]root of both sides.

0:37:43.540,0:37:47.968
And that's it. We have found[br]the ratio of the velocity to

0:37:47.968,0:37:51.658
the speed of light in terms[br]of the two times.

0:37:52.680,0:37:57.095
So. That concludes this video on[br]transformation of formula thing

0:37:57.095,0:38:02.065
to do is to treat it always as[br]though it was an equation that

0:38:02.065,0:38:03.840
you were trying to solve.

0:38:05.000,0:38:10.603
In terms of the variable that[br]you want, X equals G equals and

0:38:10.603,0:38:15.344
you just perform the steps as[br]though you were solving an

0:38:15.344,0:38:20.516
equation. If you keep that in[br]mind, then most of your problems

0:38:20.516,0:38:22.671
should be taken care of.