WEBVTT
00:00:01.260 --> 00:00:14.670
In a square π΄π΅πΆπ·, π is the point of intersection of the two diagonals, πΈ is the midpoint of π΄π΅, and πΉ is the midpoint of π΅πΆ.
00:00:15.810 --> 00:00:29.340
Three forces of magnitudes πΉ one, πΉ two, and 41 newtons are acting at π in the directions ππΈ, ππΉ, and ππ·, respectively.
00:00:30.500 --> 00:00:36.940
Given that the three forces are in equilibrium, find the values of πΉ one and πΉ two.
00:00:38.930 --> 00:00:42.890
Before starting this question, it is worth drawing a diagram.
00:00:43.820 --> 00:00:47.630
We have a square π΄π΅πΆπ· as shown.
00:00:48.550 --> 00:01:00.600
Weβre told that π is the point of intersection of the two diagonals, πΈ is the midpoint of π΄π΅, and πΉ is the midpoint of π΅πΆ.
00:01:01.650 --> 00:01:05.550
We also know that there are three forces acting at π.
00:01:06.610 --> 00:01:09.790
πΉ one acts in the direction ππΈ.
00:01:10.780 --> 00:01:13.960
πΉ two acts in the direction ππΉ.
00:01:14.830 --> 00:01:20.530
And finally, we have a 41-newton force acting in the direction ππ·.
00:01:21.860 --> 00:01:27.390
As the three forces are in equilibrium, there are several ways of solving this problem.
00:01:28.560 --> 00:01:31.840
In this question, we will use Lamiβs theorem.
00:01:32.550 --> 00:01:50.010
This states that if three forces in equilibrium are acting at a point, in this case π΄, π΅, and πΆ, where the angle between forces π΅ and πΆ is πΌ, between π΄ and πΆ is π½, and between π΄ and π΅ is πΎ.
00:01:50.790 --> 00:01:52.170
Then π΄ over sin.
00:01:52.170 --> 00:01:58.340
πΌ is equal to π΅ over sin π½, which is equal to πΆ over sin πΎ.
00:01:59.230 --> 00:02:03.920
You may also notice that this is the sine rule formula from trigonometry.
00:02:05.070 --> 00:02:12.710
In our question, the angle between the force πΉ one and πΉ two is 90 degrees as they are perpendicular.
00:02:13.860 --> 00:02:18.180
The two diagonals of a square also meet at right angles.
00:02:19.140 --> 00:02:29.410
As half of a right angle is 45 degrees, the angle between πΉ one and the 41-newton force is 135 degrees.
00:02:30.090 --> 00:02:41.850
The angle between πΉ two and the 41-newton force is also 135 degrees as angles in a circle or at a point sum to 360.
00:02:42.960 --> 00:02:57.810
Substituting these values into Lamiβs Theorem gives us 41 over sin 90 is equal to πΉ one over sin 135 which is equal to πΉ two over sin of 135.
00:02:59.140 --> 00:03:01.960
Letβs consider the first two terms.
00:03:02.870 --> 00:03:05.920
The sin of 90 degrees is equal to one.
00:03:06.800 --> 00:03:14.420
This means that 41 is equal to πΉ one over or divided by sin of 135.
00:03:15.180 --> 00:03:27.870
Multiplying both sides of this equation by sin of 135 gives us πΉ one is equal to 41 multiplied by sin 135.
00:03:28.840 --> 00:03:31.490
We could just type this into our calculator.
00:03:32.190 --> 00:03:37.980
However, we know that sin of 45 degrees is equal to one over root two.
00:03:39.360 --> 00:03:42.850
This can also be written as root two over two.
00:03:43.830 --> 00:03:51.160
If the sum of two angles equals 180 degrees, then the sin of both of these angles are equal.
00:03:51.580 --> 00:03:57.640
Therefore, the sin of 135 is also equal to root two over two.
00:03:58.790 --> 00:04:04.960
This means that πΉ one is equal to 41 multiplied by root two over two.
00:04:06.230 --> 00:04:13.790
We can, therefore, say that the force πΉ one is equal to 41 root two over two newtons.
00:04:14.740 --> 00:04:21.510
If we consider the second and third term in Lamiβs Theorem, we noticed that the denominators are the same.
00:04:21.860 --> 00:04:25.520
Theyβre both sin of 135 degrees.
00:04:26.450 --> 00:04:33.540
If two fractions are equal and their denominators are equal, this means that the numerators must also be equal.
00:04:34.550 --> 00:04:44.080
As πΉ one is equal to 41 root two over two, then πΉ two must also be equal to 41 root two over two.
00:04:45.190 --> 00:04:51.500
Both of the forces in this case are 41 root two over two newtons.