Video: Analyzing the Resolved Components of a Force

The diagram shows a body of weight 69 N suspended by 2 light, inextensible strings, ๐ด๐ถ and ๐ต๐ถ. Both strings make an angle of 37ยฐ with the horizontal. Resolve the weight of the body into two components in the direction ๐ด๐ถ and in the direction ๐ต๐ถ. Give your answers to the nearest newton.

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Video Transcript

The diagram shows a body of weight 69 newtons suspended by two light, inextensible strings, ๐ด๐ถ and ๐ต๐ถ. Both strings make an angle of 37 degrees with the horizontal. Resolve the weight of the body into two components in the direction ๐ด๐ถ and in the direction ๐ต๐ถ. Give your answers to the nearest newton.

In order to solve this problem, weโ€™re going to use Lamiโ€™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 ๐ด, ๐ต, and ๐ถ acting on a particle, making angles ๐›ผ, ๐›ฝ, and ๐›พ with each other, then ๐ด divided by sin ๐›ผ is equal to ๐ต divided by sin ๐›ฝ is equal to ๐ถ divided by sin ๐›พ.

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 ๐ด๐ถ๐ต 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 ๐ด๐ถ๐ต is 106 degrees.

As triangle ๐ด๐ต๐ถ is isosceles, the angle between ๐‘Š one and 69 is 53 degrees. And the angle between ๐‘Š two and 69 newtons is also 53 degrees. Substituting these values into Lamiโ€™s theorem gives us ๐‘Š one divided by sin 53 is equal to ๐‘Š two divided by sin 53, which is also equal to 69 divided by sin 106.

In order to calculate ๐‘Š one, the weight of the body in the direction ๐ด๐ถ, we can solve the equation ๐‘Š 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 ๐‘Š 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.

๐‘Š two, the weight of the body in the direction ๐ต๐ถ, can be solved in exactly the same way. ๐‘Š 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 ๐‘Š two is equal to 57 newtons to the nearest newton.

This method using Lamiโ€™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.

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