WEBVTT
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Resolve a force of 81 newtons into two perpendicular components, 𝐹 one and 𝐹 two, as shown in the figure.
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Give your answer correct to two decimal places.
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In order to solve this problem, we can use Lami’s theorem.
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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.
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This means that if we have three forces 𝐴, 𝐵, and 𝐶, then 𝐴 divided by sin 𝛼, the angle between force 𝐵 and force 𝐶, is equal to 𝐵 divided by sin 𝛽, which is equal to 𝐶 divided by sin 𝛾.
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Our first step in the figure shown is to work out the missing angle in the diagram.
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Well, if two forces are perpendicular to each other, then they must meet at 90 degrees.
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This means that the angle between the force 𝐹 two and the 81-newton force is 36 degrees, as 90 minus 54 is equal to 36.
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Applying Lami’s theorem gives us 𝐹 one divided by sin 36, as the 36-degree angle is between the force 𝐹 two and 81.
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This is equal to 𝐹 two divided by sin 54, as the 54-degree angle is between the force 81 newtons and 𝐹 one.
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This is also equal to 81 divided by sin 90, as the 90-degree angle is between the forces 𝐹 one and 𝐹 two.
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To calculate the force 𝐹 one we will solve 𝐹 one divided by sin 36 is equal to 81 divided by sin 90.
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Multiplying both sides of this equation by sin 36 gives us that 𝐹 one is equal to 81 divided by sin 90 multiplied by sin 36.
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This gives us a value for the force 𝐹 one of 47.61 newtons to two decimal places.
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In order to calculate the force 𝐹 two, we will solve the equation 𝐹 two divided by sin 54 is equal to 81 divided by sin 90.
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Multiplying both sides of this equation by sin 54 gives us 𝐹 two is equal to 81 divided by sin 90 multiplied by sin 54.
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This is equal to 65.32 newtons.
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Therefore, 𝐹 two is equal to 65.32 newtons to two decimal places.
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The two perpendicular components 𝐹 one and 𝐹 two, shown in the figure, are 47.61 newtons and 65.32 newtons.