Center of mass: A 2.00-m rod of negligible mass connects two very small objects at its ends. The mass of one object is 1.00 kg and the mass of the other is unknown. The center of mass of this system is on the rod a distance 1.60 m from the 1.00-kg mass object. What is the mass of the other object?

Question

Quizplus · Accepted Answer

We can use the formula for the center of mass:
$R_{CM} = \dfrac{m_1r_1 + m_2r_2}{m_1 + m_2}$
Where $m_1$ is the mass of the known object (1.00 kg), $r_1$ is the distance from the center of mass to the known object (1.60 m), $m_2$ is the mass of the unknown object, and $r_2$ is the distance from the center of mass to the unknown object (which we do not yet know).
Plugging in the given values:
$1.60 m = \dfrac{(1.00 kg)(1.60 m) + (m_2)(r_2)}{(1.00 kg + m_2)}$
Simplifying:
$1.60 m (1.00 kg + m_2) = (1.00 kg)(1.60 m) + (m_2)(r_2)$
$1.60 kg \cdot m = 1.60 kg + r_2 m_2$
$r_2 = \dfrac{1.60 kg \cdot m - 1.60 kg}{m_2}$
$r_2 = 1.60 m - \dfrac{1.60 kg}{m_2}$
We know that $r_2$ must be positive (since it is the distance from the center of mass to the unknown object), so we can set up an inequality:
$1.60 m - \dfrac{1.60 kg}{m_2} > 0$
Solving for $m_2$:
$m_2 > \dfrac{1.60 kg}{1.60 m}$
$m_2 > 1.00 kg$
The only answer choice greater than 1.00 kg is C, which gives a mass of 4.00 kg. We can verify that this satisfies the inequality:
$r_2 = 1.60 m - \dfrac{1.60 kg}{4.00 kg} = 1.20 m > 0$ 
Therefore, the answer is C: 4.00 kg.

Center of Mass: a 2