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Derive an expression for the center of mass of two point masses. |
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Answer» Solution :(i) Consider the point masses `m_(1) and m_(2)` which are positioned as `x_(1) and x_(2)` along X-axis. The center of mass can be found in this system in THREE ways as follows (a) When the masses are on positive X-axis : (i) The ORIGIN is taken arbitrarily as shown in figure.  (II) The center of mass along X-axis is, `x_(CM) = (m_(1)x_(1)+m_(2)x_(2))/(m_(1)+m_(2))` (b) When the origin coincides with any one of the masses : (i) If the orging coincide with mass `m_(1)` as shown in figure, its position `x_(1) = 0`  (ii) The center of mass along X-axis is, `x_(CM) = (m_(1)(0) + m_(2) x_(2))/(m_(1)+m_(2)) = (m_(2)x_(2))/(m_(1)+m_(2))` (c) When the origin coincides with the center of mass itself : (i) If the origin coincide with center of mass as shown in figure, `X_(CM) = 0`. Hence, the position `x_(1)` is negative.  (ii) The center of mass along X-axis is, `0 = (m_(1)(-x_(i))+m_(2)x_(2))/(m_(1)+m_(2))` `m_(1)x_(1) = m_(2)x_(2)` (III) The above equation is known as principle of moments. |
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