A body of weight 90 kilogram-weight is placed on a smooth plane that is inclined at 30 degrees to the horizontal. If the body is held in equilibrium by means of a force 𝐹 that acts at an angle of 30 degrees above the plane, determine the magnitudes of 𝐹 and 𝑟, where 𝑟 is the reaction of the plane on the body.
We will begin by sketching a diagram that models the situation. A body of weight 90 kilogram-weight is placed on a smooth plane. We are told that the plane is inclined at an angle of 30 degrees. The body is held in equilibrium by a force 𝐹. And this acts at an angle 30 degrees above the plane. We are asked to calculate the magnitude of this force 𝐹 together with 𝑟, which is the reaction of the plane on the body. The reaction force will act in the direction perpendicular to the plane.
In this question, we have three forces acting at a point. Since the body is in equilibrium, we can use Lami’s theorem to help solve it. This states that when three forces acting on a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. If we have three forces 𝐴, 𝐵, and 𝐶, then 𝐴 over sin 𝛼 is equal to 𝐵 over sin 𝛽, which is equal to 𝐶 over sin 𝛾, where 𝛼 is the angle between forces 𝐵 and 𝐶, 𝛽 is the angle between the forces 𝐴 and 𝐶, and 𝛾 is the angle between the forces 𝐴 and 𝐵.
In our diagram, we will begin by calculating the angles between the three forces. The angle between the force 𝐹 and the reaction force is 60 degrees, as 90 minus 30 is equal to 60. This is the angle that is opposite the weight force. The angle between the reaction force and the weight force is 150 degrees, as this is 90 degrees plus 60 degrees. This angle is opposite the force 𝐹. Finally, the angle between the weight force and the force 𝐹 is also 150 degrees, as angles at a point must sum to 360 degrees. This angle is opposite the reaction force 𝑟.
We can now substitute these values into Lami’s theorem. We have 𝐹 over the sin of 150 degrees is equal to 𝑟 over the sin of 150 degrees, which is equal to 90 over sin 60 degrees. The sin of 150 degrees is one-half. And since dividing by a half is the same as multiplying by two, the first two parts of our equation simplify to two 𝐹 and two 𝑟. The sin of 60 degrees is equal to root three over two. 90 divided by root three over two is 180 over root three. We can rationalize this by multiplying the numerator and denominator by root three. This gives us 180 root three over three, which in turn simplifies to 60 root three. Two 𝐹 and two 𝑟 are both equal to 60 root three. Dividing through by two, we see that 𝐹 is equal to 𝑟, which is equal to 30 root three.
We can therefore conclude that the force 𝐹 is equal to 30 root three kilogram-weight and the reaction force 𝑟 is also equal to 30 root three kilogram-weight.