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Hi there!
We can use a summation of torques to solve.
Recall the equation for torque:
[tex]\large\boxed{\Sigma \tau = rF}[/tex]
r = distance from fulcrum (balance point)
F = force (in this instance, weight, N)
We can set the fulcrum to be the balance point of 30 cm.
Thus:
Meter ruler:
Center of mass at 48 cm ⇒ 48 - 30 = 18 cm
Object:
At 6cm ⇒ 30 - 6 = 24 cm
For the ruler to be balanced:
[tex]\large\boxed{\Sigma \tau_{cc} = \Sigma \tau_{ccw}}[/tex]
Thus:
[tex]M_Rg(18) = 60g(24)\\M_R = \frac{60(24)}{18} = \boxed{80 g}[/tex]
The mass of the ruler is 80 grams.
If the body were moved to 13 cm:
B (balance point) - 13 = distance of object
48 - B = distance from ruler center of mass to balance point
[tex]80g(48 - B) = 60g(B - 13)\\\\3840 - 80B = 60B - 780\\\\4620 = 140B\\\boxed{B = 33 cm}[/tex]
The new balance point would be 33cm from the zero end.