How is the Coordinate of M1 Determined in a Dumbbell System?

[rsaad]

Homework Statement

Attached is the picture of a dumbbell. I do not understand how the coordinate of M₁ is
M₂ R/ (M₁ + M₂)

It is not an assignment question but an example from a book.

 

[ehild]

The origin is at the centre of mass. You certainly know how to determine the position of the CM?

ehild

 

[rsaad]

CM= (R₁m₁+R₂M₂)/M₁+M₂
because CM=0, I get M₁ R₁ = -R₂ M₂
which further gives me
R₁= RM₂ / (M₂-M₁)

=|

 

[ehild]

CM= (R₁m₁+R₂M₂)/M₁+M₂
because CM=0, I get M₁ R₁ = -R₂ M₂
which further gives me
R₁= RM₂ / (M₂-M₁)

=|

What do you mean on R₁ and R₂? If they are coordinates of the spheres with respect to the CM, the formula is correct, but use parentheses. (R₁m₁+R₂M₂)/(M₁+M₂)=0

How are R1 and R2 related to the distance R between the spheres?

ehild

 

[rsaad]

R= R_1 + R_2
where R_i is the distance to M_i from CM.

 

[rsaad]

In the book R_1 = R M2/ (M1 + M2) but mine is R1= R M2 / (M2-M1)

 

[ehild]

If x1, x2 are coordinates of m1, m2, respectively, the x coordinate of the CM is (x1m1+
x2m2)/(m1+m2).
If they are distances from the CM, the coordinate of m2 is negative: m1R1-M2R2=0

ehild