The center of mass of a set of points p1, . . , pn is defined as the point mn = (p1 + · · · + pn)/n. (a) for a tetrahedron p1p2p3p4, consider the point dividing the segment m3p4 in the ratio 1: 3, where m3 is the center of mass of the vertices p1p2p3. show that this point is the center of mass of the tetrahedron vertices, and that its location is independent of the order in which the vertices are taken. (b) two edges of the tetrahed

solution:

a)

M₃= P₁+P₂+P₃/3

Now the let point A be center of mass tetrahedran then,

Acm = P₁+P₂+P₃+P₄/4

4Acm =P₁+P₂+P₃+P₄

P₁+P₂+P₃= 3M₃

4Acm = 3M₃ + p₄   1

Let a point o be the point that devide line  

3M₃p₄ into radio 1:3

i.e M₃o/o P₄  = 1/3

so, center of mass of this point will be,

0 = xcm = 3M₃o + o P₄ /3+1

40 = 3M₃o + o P₄   2

From 1 and 2 ,

We can see that o = Acm

Hence, we have considered arbitrary points.

xcm = P₁+P₂+P₃+P₄/4

now, the mid point of P₁P₂ = P₁+P₂/2

         x P₁P₂ = P₁+P₂/2

         x P₃P₄ = x P₃ + P₄/2

now center of mass of  x P₁P₂ & x P₃P₄  

  = XP₁P₂P₃P₄ = x P₁P₂ + x P₃P₄/2

XP₁P₂P₃P₄  =  x P₁+P₂+ P₃+P₄ = Xcm

Hence it is proved that line joining midpoint passes through c.m

idk

step-by-step explanation:

idk

d

step-by-step explanation:

hello and welcome!

so, when we see the words "proportional" along with a table, we need to know that we're looking numbers that divide by the same thing.

looking at this table, we simply have to do a trial and error, and divide to see which numbers have the same quotient. if you look at choice a & c, they both have equal quotients for three of the numbers, with one exception, so those can't be the answer. looking at b, there is no similarity, so that can't be it. then, we see d; if you divide all the numbers in the table, you'll see that they all divide by two, which is a proportional relationship, and our answer!

also, always remember that the x will always be the number that goes into y, and somewhere behind this table, is a formula. for this specific table, the formula would be:

since our x-value is being divided by two to get the y-value. and if they were to ask for more numbers in the series, this is the formula you would use to find them.