Question Video: Resolving a Force into Two Nonperpendicular Components

A force of magnitude 96 N acts vertically downward. It is resolved into two components as shown in the diagram. Find the magnitude of ๐นโ‚ and ๐นโ‚‚ giving values to two decimal places.

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Video Transcript

A force of magnitude 96 newtons acts vertically downward. It is resolved into two components as shown in the diagram. Find the magnitude of ๐น one and ๐น two, giving values to two decimal places.

In order to solve this problem, we will use Lamiโ€™s theorem. This states that if three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

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. This is because angle ๐›ผ is between forces ๐ต and ๐ถ. Angle ๐›ฝ is between forces ๐ด and ๐ถ. And finally, angle ๐›พ is between forces ๐ด and ๐ต.

In our example, the angle between force ๐น one and 96 newtons is 90 degrees. The angle between ๐น two and 96 newtons is 42 degrees. And the angle between forces ๐น one and ๐น two is 132 degrees. This means that ๐น one divided by sin 42 is equal to ๐น two divided by sin 90, which is equal to 96 divided by sin 132.

If we consider the forces ๐น one and 96 newtons, we can rearrange the equation so that ๐น one is equal to 96 divided by sin 132 multiplied by sin of 42. This gives us an answer for ๐น one of 86.44 newtons. When we consider the two forces ๐น two and 96 newtons, we can rearrange this equation to give us ๐น two is equal to 96 divided by sin 132 multiplied by sin 90. This gives us a value of ๐น two of 129.18 newtons.

Using Lamiโ€™s theorem, weโ€™ve proved that the force ๐น one is equal to 86.44 newtons and ๐น two is equal to 129.18 newtons.

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