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A body of weight 90 kilogram-weight is placed on a smooth plane that is inclined at 30 degrees to the horizontal.
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If the body is held in equilibrium by means of a force πΉ that acts at an angle of 30 degrees above the plane, determine the magnitudes of πΉ and π, where π is the reaction of the plane on the body.
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We will begin by sketching a diagram that models the situation.
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A body of weight 90 kilogram-weight is placed on a smooth plane.
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We are told that the plane is inclined at an angle of 30 degrees.
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The body is held in equilibrium by a force πΉ.
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And this acts at an angle 30 degrees above the plane.
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We are asked to calculate the magnitude of this force πΉ together with π, which is the reaction of the plane on the body.
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The reaction force will act in the direction perpendicular to the plane.
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In this question, we have three forces acting at a point.
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Since the body is in equilibrium, we can use Lamiβs theorem to help solve it.
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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.
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If we have three forces π΄, π΅, and πΆ, then π΄ over sin πΌ is equal to π΅ over sin π½, which is equal to πΆ over sin πΎ, where πΌ is the angle between forces π΅ and πΆ, π½ is the angle between the forces π΄ and πΆ, and πΎ is the angle between the forces π΄ and π΅.
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In our diagram, we will begin by calculating the angles between the three forces.
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The angle between the force πΉ and the reaction force is 60 degrees, as 90 minus 30 is equal to 60.
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This is the angle that is opposite the weight force.
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The angle between the reaction force and the weight force is 150 degrees, as this is 90 degrees plus 60 degrees.
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This angle is opposite the force πΉ.
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Finally, the angle between the weight force and the force πΉ is also 150 degrees, as angles at a point must sum to 360 degrees.
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This angle is opposite the reaction force π.
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We can now substitute these values into Lamiβs theorem.
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We have πΉ over the sin of 150 degrees is equal to π over the sin of 150 degrees, which is equal to 90 over sin 60 degrees.
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The sin of 150 degrees is one-half.
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And since dividing by a half is the same as multiplying by two, the first two parts of our equation simplify to two πΉ and two π.
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The sin of 60 degrees is equal to root three over two.
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90 divided by root three over two is 180 over root three.
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We can rationalize this by multiplying the numerator and denominator by root three.
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This gives us 180 root three over three, which in turn simplifies to 60 root three.
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Two πΉ and two π are both equal to 60 root three.
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Dividing through by two, we see that πΉ is equal to π, which is equal to 30 root three.
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We can therefore conclude that the force πΉ is equal to 30 root three kilogram-weight and the reaction force π is also equal to 30 root three kilogram-weight.