A body weighing 𝑊 newtons is placed on a smooth plane inclined at 45 degrees to the horizontal. If it is kept in equilibrium under the action of a horizontal force of magnitude 33 newtons, find the weight of the body 𝑊 and the reaction of the plane 𝑅.
In order to answer any question like this, it is worth drawing a diagram first. The body rests on a smooth plane inclined at 45 degrees. It has a weight of 𝑊 newtons. The reaction force, 𝑅 newtons, will always be perpendicular to the plane. There is a horizontal force of 33 newtons keeping the body in equilibrium. There are lots of methods of solving this problem. In this case, we’ll use Lami’s theorem.
This states that if we have three forces acting at a point, in this case 𝐴, 𝐵, and 𝐶, and if the angle between forces 𝐵 and 𝐶 is 𝛼, between 𝐴 and 𝐶 is 𝛽, and between 𝐴 and 𝐵 is 𝛾, then 𝐴 over sin 𝛼 is equal to 𝐵 over sin 𝛽, which is equal to 𝐶 over sin 𝛾. You may also remember this formula as the sine rule in trigonometry.
In our question, we have three forces acting at the centre of the body, a vertical force downwards, 𝑊 newtons, a horizontal force, 33 newtons, and a reaction force, 𝑅 newtons. As the weight and the 33-newton force are perpendicular, the angle between them is 90 degrees. Using our knowledge of z angles or alternate angles, we can see a 45-degree angle as shown. As 90 plus 45 is equal to 135, the angle between the reaction force and the weight is 135 degrees. As angles in a circle or at a point sum to 360, the angle between the reaction force and the 33-newton force is also 135 degrees.
Focusing in on this point gives the following diagram. Substituting these values into Lami’s theorem gives us 𝑅 over sin 90 is equal to 33 over sin 135, which is equal to 𝑊 over sin of 135. If we consider the last two terms, we have 33 over sin 135 is equal to 𝑊 over sin 135. As the denominators are the same, the numerators must also be the same. This means that 𝑊, the weight of the body, is 33 newtons. Let’s now consider the first two terms. 𝑅 over sin 90 is equal to 33 over sin 135. We know that sin of 90 is equal to one. This means that 𝑅 is equal to 33 over sin of 135. Sin of 45 degrees is equal to one over root two.
From our sine graph or CAST diagram, this means that sin of 135 is also equal to one over root two. We can therefore say that 𝑅 is equal to 33 divided by one over root two. Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. This means that 𝑅 is equal to 33 root two. We can therefore conclude that the weight of the body is 33 newtons and the reaction force is 33 root two newtons.