WEBVTT
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The figure shows a system of point masses placed at the vertices of a triangle.
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The mass placed at each point is detailed in the table.
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Determine the coordinates of the centre of gravity of the system.
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Now, because the shape weβve been given isnβt symmetrical, weβre going to need to be a little bit careful about how we determine the centre of mass of our object.
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Now, if weβre given a system of π particles of masses π one, π two, and so on up to ππ, whose position vectors are π one, π two all the way up to π π, respectively.
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The centre of mass of the system is the point with position vector π and is equal to the sum from π equals one to π over π sub π times π sub π all over capital π, where thatβs the total mass of the system.
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In reality though, it can be much easier, especially in two dimensions, to just fit this into the π₯-direction and the π¦-direction.
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The π₯-coordinate of the centre of mass of a system of π particles π₯ sub πΆ is π one π₯ one plus π two π₯ two all the way up to π π π₯ π over π one plus π two all the way up to π π.
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And the π¦-coordinate of the centre of mass π¦ sub πΆ is π one π¦ one plus π two π¦ two all the way up to π π π¦ π all over the total sum of the masses.
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Now here, π₯ one, π₯ two, and π₯ π, π¦ one, π¦ two, and π¦ π are the π₯- and π¦-coordinates, respectively, of each of the individual particles.
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Weβll begin by dealing with the π₯-coordinate of the centre of mass.
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π one π₯ one is the mass of π΄ times the distance that π΄ is from the origin.
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So thatβs 13 times zero.
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π΅ is six times zero.
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Again, the horizontal distance from the origin is zero.
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And for πΆ, it is 15 times six.
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This is all over the sum of their masses.
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On the top of our fraction, this simplifies to 90 and, on the bottom, we get 34.
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This simplifies to 47 over 17.
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Weβre going to repeat this process for the π¦-coordinate.
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π one π¦ one is 13 times zero.
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Remember, π΄ is at the origin, so itβs zero units away in the vertical direction.
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This time, π΅ is eight centimetres away from the origin in the vertical direction, and πΆ is zero units away in the vertical direction.
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Once again, this is all over the sum of their masses.
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And this gives us 48 over 34.
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48 over 34 simplifies to 24 over 17.
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And so we found the coordinates of the centre of mass or centre of gravity of our system.
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They are 47 17ths and 24 17ths.