Video Transcript
A body weighing 85 newtons is placed on a smooth plane inclined at 45 degrees to the horizontal. The body is kept in equilibrium by an inextensible string fixed to a point on a vertical wall at the top of the slope. Given that the tension in the string is of magnitude 62 newtons, find the measure of the angle 𝜃 the string makes with the horizontal, giving your answer to the nearest minute, and the magnitude of the reaction 𝑅 of the plane on the body, stating your answer to the nearest two decimal places.
We can see from the diagram that we have a body of weight 85 newtons on a plane inclined at 45 degrees to the horizontal. The body is kept in equilibrium by a string with tension 62 newtons. And we are told this string makes an angle of 𝜃 degrees with the horizontal. The reaction force 𝑅 acts perpendicular to the plane.
We therefore have three forces acting at a point. And we can calculate the unknowns by firstly using Lami’s theorem. This states that when 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. If the three forces are 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 forces 𝐴 and 𝐶. And 𝛾 is the angle between the forces 𝐴 and 𝐵. In this question, we will let the angles 𝛼, 𝛽, and 𝛾 be as shown.
Recalling that the aim of this question is to calculate the angle 𝜃 and reaction force 𝑅, we will now clear some space for our working. The angle 𝛽 between the reaction force and the 85-newton force is equal to 90 degrees plus 45 degrees. This is equal to 135 degrees. Adding this to our diagram, we can now substitute this angle together with our three forces into Lami’s theorem. We have 𝑅 over sin 𝛼 is equal to 62 over sin of 135 degrees, which is equal to 85 over sin 𝛾. The first expression has two unknowns. We will therefore use the second and third expressions to calculate angle 𝛾.
Taking the reciprocal of both of these fractions, we have sin 𝛾 over 85 is equal to sin of 135 degrees over 62. We can then multiply through by 85. sin 𝛾 is therefore equal to 0.9694 and so on. Next, we take the inverse sine of both sides of this equation. This gives us 𝛾 is equal to 75.79 and so on degrees. For accuracy, at this stage, we will not round our answer. Since 𝛼, 𝛽, and 𝛾 sum to 360 degrees, we can use this to calculate angle 𝛼. We subtract 135 degrees and 75.79 degrees from 360 degrees. This gives us 𝛼 is equal to 149.20 and so on degrees.
We are now in a position to calculate the value of angle 𝜃. We recall that this is the angle that the string makes with the horizontal. 𝜃 is therefore equal to 149.20 and so on minus 90 degrees. Using the more accurate answer from our calculator, we have 𝜃 is equal to 59.20578 and so on degrees. We were asked to give our answer in degrees and minutes. We can do this either by multiplying the decimal part of our answer by 60 or by using the degrees, minutes, and seconds button on our calculator. Either way, we get an answer of 59 degrees and 12 minutes to the nearest minute. The measure of the angle that the string makes with the horizontal is 59 degrees and 12 minutes.
Since we now know the value of angle 𝛼, we can go back to Lami’s theorem to calculate the reaction force 𝑅. Once again, we will use the exact answer. We have 𝑅 over the sin of 149.20 and so on degrees, which is equal to 62 over sin of 135 degrees. Multiplying through by the sin of angle 𝛼 gives us 𝑅 is equal to 44.8889 and so on. We were asked to give this value correct to two decimal places. And it is therefore equal to 44.89. The magnitude of the reaction 𝑅 of the plane on the body is 44.89 newtons.
