The diagram shows a body of weight 69 newtons suspended by two light, inextensible strings, 𝐴𝐶 and 𝐵𝐶. Both strings make an angle of 37 degrees with the horizontal. Resolve the weight of the body into two components in the direction 𝐴𝐶 and in the direction 𝐵𝐶. Give your answers to the nearest newton.
In order to solve this problem, we’re going to use Lami’s theorem. This states that if three forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces.
If we can set up the three forces 𝐴, 𝐵, and 𝐶 acting on a particle, making angles 𝛼, 𝛽, and 𝛾 with each other, then 𝐴 divided by sin 𝛼 is equal to 𝐵 divided by sin 𝛽 is equal to 𝐶 divided by sin 𝛾.
Our first step is to work out the missing angles in the diagram. Well, angles in a triangle add up to 180 degrees. This means that angle 𝐴𝐶𝐵 can be calculated by subtracting 37 plus 37 from 180. Well, 37 plus 37 is 74. And 180 minus 74 is 106 degrees. This means that angle 𝐴𝐶𝐵 is 106 degrees.
As triangle 𝐴𝐵𝐶 is isosceles, the angle between 𝑊 one and 69 is 53 degrees. And the angle between 𝑊 two and 69 newtons is also 53 degrees. Substituting these values into Lami’s theorem gives us 𝑊 one divided by sin 53 is equal to 𝑊 two divided by sin 53, which is also equal to 69 divided by sin 106.
In order to calculate 𝑊 one, the weight of the body in the direction 𝐴𝐶, we can solve the equation 𝑊 one divided by sin 53 is equal to 69 divided by sin 106. Multiplying both sides of the equation by sin 53 gives us that 𝑊 one is equal to 69 divided by sin 106 multiplied by sin 53. This gives us an answer of 57.33, or 57 newtons to the nearest newton.
𝑊 two, the weight of the body in the direction 𝐵𝐶, can be solved in exactly the same way. 𝑊 two divided by sin 53 is equal to 69 divided by sin 106. As this is the same equation we solved earlier, we can also say that 𝑊 two is equal to 57 newtons to the nearest newton.
This method using Lami’s theorem can be used anytime we have three forces acting at a point that are in equilibrium. They can be used to work out the missing forces or the missing angles.