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Two children of mass 20.0 kg and 30.0 kg sit balanced on a seesaw with the pivot point located at the center of the seesaw. If the children are separated by a distance of 3.00 m, at what distance from the pivot point is the small child sitting in order to maintain the balance?
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$1.8 \textrm{ m}$

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OpenStax College Physics Solution, Chapter 9, Problem 7 (Problems & Exercises) (2:13)

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Video Transcript

This is College Physics Answers with Shaun Dychko. We know that a boy of mass 20 kilograms is sitting on one side of the pivot on the seesaw and a girl of mass 30 kilograms is sitting on the other side and they are meant to be in static equilibrium. So that means that they are exerting the same magnitude torques in opposite directions and the total distance between them is three meters. So that means the level arm of boy one which is <i>r1</i> plus the level arm of the girl which is <i>r2</i> has to equal three. So, we’ll use that in a second. First of all, let’s just say that the counter-clockwise torque has to equal the clockwise torque. And so the counter-clockwise torque is due to the boy, so that’s its weight multiplied by its level arm, so that’s <i>m1 g</i> times <i>r1</i> and then for the girl, it’s <i>m2 g</i> times <i>r2</i>. The weight is perpendicular to the level arm already so we don’t need to be concerned with angles here, or you could just go sine of 90 degrees which should be the number one. So then we use this fact here to replace the <i>r2</i> in this equation in order to have an equation consisting of only one variable <i>r1</i> which we’ll then solve for. So we replace <i>r2</i> with three meters minus <i>r1</i>, and then we solve for <i>r1</i>. So we multiply through by <i>m2</i>, then we get <i>m2</i> times <i>r1</i> here with a minus in front of it and then we add that to both sides, then we get this line here. We have <i>m1 r1</i> plus <i>m2 r2</i> equals three <i>m2</i>. And then factor out the <i>r1</i>, which is a common factor among these two terms, and then divide by the bracket <i>m1</i> plus <i>m2</i> that we get, and then you end up with <i>r1</i> is three times <i>m2</i> divided by <i>m1</i> plus <i>m2</i>. So that’s three meters times 30 kilograms divided by 20 kilograms plus 30 kilograms, which is 1.8 meters is the distance from the pivot that the boy needs to sit in order to create a torque that balances the torque exerted by the girl in the opposite direction.