WEBVTT
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Three forces, five π plus 10π newtons, ππ minus five π newtons, and 15π plus π plus seven π newtons, act on a particle.
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Given that the resultant of the forces is 18π plus 19π newtons, what are the values of π and π?
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Weβve been given the vector representation of three forces in newtons acting on a single particle.
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Weβve also been given their resultant.
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Thatβs the total of these three forces.
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In order to work out the values of π and π then, weβll need to use this information to set up equations in terms of π and π.
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And since the resultant is the sum total of the three forces, we begin by adding them.
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Thatβs five π plus 10π plus ππ minus five π plus 15π plus π plus seven π.
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And to add vectors, we add their individual components.
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So weβll add their horizontal components.
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Thatβs five, π, and 15.
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And weβll add their vertical components.
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Thatβs 10, negative five, and π plus seven.
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Five plus π plus 15 simplifies to 20 plus π.
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And 10 minus five plus π plus seven simplifies to 12 plus π.
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So we have the horizontal component to be 20 plus π and the vertical component to be 12 plus π.
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Now we know that the actual horizontal component of the resultant of the forces is 18.
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And the actual vertical component is 19.
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We can say then that 20 plus π must be equal to 18.
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And 12 plus π must therefore be equal to 19.
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And then, weβll solve these equations for π and π.
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To solve this first equation, letβs subtract 20 from both sides.
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We get π is equal to negative two.
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And to solve the second equation, weβll subtract 12 from both sides.
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And we get π is equal to seven.
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And weβve set out what we needed to do.
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We found the values of π and π.
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π is equal to negative two.
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And π is equal to seven.