WEBVTT
00:00:01.640 --> 00:00:07.640
A body of weight 𝑤 is attached to a wall by a string of length 25 centimeters.
00:00:07.640 --> 00:00:19.160
It is held in equilibrium by the effect of a horizontal force of magnitude 93 gram-weight that keeps the body 15 centimeters away from the wall.
00:00:20.080 --> 00:00:21.960
Determine 𝑇 and 𝑤.
00:00:23.560 --> 00:00:28.640
In order to answer this question, we will consider a triangle of forces in equilibrium.
00:00:29.640 --> 00:00:32.320
There are three forces acting at point 𝐶.
00:00:32.960 --> 00:00:42.000
We have the weight of the body 𝑤, a horizontal force of magnitude 93 gram-weight, and the tension force in the string.
00:00:43.480 --> 00:00:47.360
Since the system is in equilibrium, there is no resultant force.
00:00:48.040 --> 00:00:58.040
And to form a triangle of forces with zero resultant, the magnitudes of the forces must be in the same ratio as the lengths of the sides of the triangle.
00:00:59.440 --> 00:01:04.000
We will therefore begin by calculating the missing length of the triangle, side 𝐴𝐵.
00:01:04.760 --> 00:01:18.280
The Pythagorean theorem states that 𝐴 squared plus 𝐵 squared is equal to 𝐶 squared, where 𝐶 is the length of the longest side of a right triangle and 𝐴 and 𝐵 are the lengths of the shorter sides.
00:01:19.280 --> 00:01:27.040
Substituting the values from the diagram, we have 𝑥 squared plus 15 squared is equal to 25 squared.
00:01:28.160 --> 00:01:35.640
This simplifies to 𝑥 squared plus 225 is equal to 625.
00:01:36.400 --> 00:01:43.480
Subtracting 225 from both sides, we have 𝑥 squared is equal to 400.
00:01:43.480 --> 00:01:46.840
We can then square root both sides of the equation.
00:01:47.320 --> 00:01:52.320
And since 𝑥 must be positive, we have 𝑥 is equal to 20.
00:01:53.040 --> 00:01:56.920
Side length 𝐴𝐵 is equal to 20 centimeters.
00:01:58.440 --> 00:02:05.520
We might also notice that the side lengths of our triangle are in the ratio 15:20:25.
00:02:06.120 --> 00:02:10.360
This simplifies to three, four, five.
00:02:11.160 --> 00:02:15.680
And we have a three-four-five Pythagorean triple.
00:02:16.720 --> 00:02:23.640
As already mentioned, the magnitudes of the forces must be in the same ratio as the lengths of the sides of the triangle.
00:02:24.440 --> 00:02:33.120
This means that 𝑇 over 25 is equal to 93 over 15, which is equal to 𝑤 over 20.
00:02:34.320 --> 00:02:44.080
Dividing each of the denominators by five, we have 𝑇 over five is equal to 93 over three, which is equal to 𝑤 over four.
00:02:45.080 --> 00:02:51.640
We can now solve the equation 𝑇 over five is equal to 93 over three to calculate the value of 𝑇.
00:02:52.800 --> 00:02:56.280
93 divided by three is 31.
00:02:56.920 --> 00:03:05.120
And multiplying through by five, we have 𝑇 is equal to 31 multiplied by five, which is equal to 155.
00:03:05.880 --> 00:03:11.280
The tension in the string is therefore equal to 155 gram-weight.
00:03:12.400 --> 00:03:16.120
We can repeat this process to calculate the weight of the body 𝑤.
00:03:16.720 --> 00:03:19.440
This is equal to 31 multiplied by four.
00:03:19.840 --> 00:03:24.280
The weight of the body is therefore equal to 124 gram-weight.
00:03:24.840 --> 00:03:28.960
And we now have values of 𝑇 and 𝑤 as required.
00:03:29.440 --> 00:03:37.000
𝑇 is equal to 155 gram-weight, and 𝑤 is equal to 124 gram-weight.