### Video Transcript

A body weighing 72 newtons is placed on a plane that is inclined at 45 degrees to the horizontal. Resolve its weight into two components ๐น one and ๐น two, where ๐น one is the component in the direction of the plane and ๐น two is the component normal to the plane.

The forces ๐น one and ๐น two are perpendicular to each other. Therefore, the angle between them is 90 degrees. The angle between the force ๐น one and the weight 72 newtons is 45 degrees and the angle between force ๐น two and 72 newtons is also 45 degrees. This question can be solved using Lamiโs theorem, which 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.

๐ด divided by sin ๐ผ is equal to ๐ต divided by sin ๐ฝ which is equal to ๐ถ divide by sin ๐พ, where the angle ๐ผ is between the forces ๐ต and ๐ถ. ๐ฝ is the angle between forces ๐ด and ๐ถ and ๐พ is the angle between forces ๐ด and ๐ต. Substituting our values into Lamiโs theorem gives us ๐น one divided by sin 45 is equal to ๐น two divided by sin 45, which is equal to 72 divided by sin 90.

We can rearrange the equation circled to give us ๐น one is equal to 72 divided by sin 90 multiplied by sin 45. This gives us a value of ๐น one of 36 root two or 50.91 to two decimal places. This means that the force ๐น one in the direction of the plane is 36 root two newtons.

We can solve to calculate ๐น two in a similar way. ๐น two is equal to 72 divided by sin 90 multiplied by sin 45. This means that ๐น two is also equal to 36 root two. The component of the force normal to the plane is 36 root two newtons.

We can go one step further here by saying that two perpendicular forces ๐น one and ๐น two will be equal to each other when the body is inclined at 45 degrees to the horizontal. This is because 45 degrees bisects the perpendicular forces.