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(A) For an object to undergo pure translational motion without rotation,the external force must be applied at its center of mass $(CM)$.Let the mass o

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$\frac{4}{3}l$

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(A) For an object to undergo pure translational motion without rotation,the external force must be applied at its center of mass $(CM)$.Let the mass of the horizontal rod $AB$ of length $l$ be $m$. Then the mass of the vertical rod $CD$ of length $2l$ will be $2m$.The center of mass of rod $AB$ is at point $D$ (midpoint of $AB$),and the center of mass of rod $CD$ is at its midpoint,which is at a distance $l$ from $D$ and $l$ from $C$.Let point $C$ be the origin $(0,0)$.Coordinates of the center of mass of rod $CD$ $(m_1 = 2m)$: $(0, l)$.Coordinates of the center of mass of rod $AB$ $(m_2 = m)$: $(0, 2l)$.The $y$-coordinate of the center of mass of the system is given by:$Y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2}$$Y_{cm} = \frac{(2m)(l) + (m)(2l)}{2m + m}$$Y_{cm} = \frac{2ml + 2ml}{3m} = \frac{4ml}{3m} = \frac{4}{3}l$Thus,the force must be applied at a distance of $\frac{4}{3}l$ from point $C$.

$A$ $T$-shaped object is placed on a smooth surface as shown in the figure. $A$ force $\vec{F}$ is applied at point $P$ in a direction parallel to $AB$ such that the object undergoes pure translational motion without rotation. Find the position of point $P$ relative to point $C$.

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