Question

Two bodies of mass \[1kg\]and \[3kg\]have position vectors as\[\hat i + 2\hat j + \hat k\] and\[\hat i + 2\hat j + \hat k\] respectively. The center of mass of this system has a position vector:
A. \[ - \hat i + 2\hat j + \hat k\]
B. \[ - 2\hat i + 2\hat k\]
C. \[ - 2\hat i - \hat j + \hat k\]
D. \[2\hat i - \hat j - 2\hat k\]

Answer 1

Hint To calculate the center of mass of any no. of bodies we have the formula as ,\[P = \dfrac{{{m_1}{p_1} + {m_2}{p_2}}}{{{m_1} + {m_2}}}\]
This is for two bodies but in the same format we can calculate the center of mass of any number of bodies by adding mass in denominator and product in numerator.

Complete step by step answer As in the given question we are given with one of mass as, \[{m_1} = 1\]
And we also given with mass of second body as, \[{m_2} = 3\]
And we know the position vector of first body as, \[{p_1} = \hat i + 2\hat j + \hat k\]
And we also given with position vector of second mass as, \[{p_2} = \hat i + 2\hat j + \hat k\]
And we know the formula to calculate center of mass as, \[P = \dfrac{{{m_1}{p_1} + {m_2}{p_2}}}{{{m_1} + {m_2}}}\]
\[P = \dfrac{{1 \times \left( {\hat i + 2\hat j + \hat k} \right) + 3 \times \left( {\hat i + 2\hat j + \hat k} \right)}}{{1 + 3}}\]
\[P = \dfrac{{4\hat i + 8\hat j + 4\hat k}}{{1 + 3}}\]
\[P = \hat i + 2\hat j + \hat k\]
So we get the position vector of the center of mass as \[P = \hat i + 2\hat j + \hat k\].

Additional information Center of mass of any body is at center where all the weight of body is referred, example center of mass of any rigid body is at the centroid of the body or it can also be referred to as different objects where relative weight of different bodies can be referred. This point is also useful to solve some problems because we assume force is acting at the point of center of mass and if we push any rigid body except the point object and apply force at center of mass then the object will never rotate with respect to any axis and will only move in the direction of force.

Note If in any question we are given to find center of gravity then don’t mix it with center of mass because in uniform gravitational field center of gravity coincides with center of mass else it doesn't.