A T-shaped object with dimensions shown in the figure is lying on a smooth floor. A force $\to F$ is applied at the point P parallel to AB, such that the object has only the translational motion without rotation. Find the location of P with respect to C.

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Solution

Given: A T-shaped object with dimensions shown in the figure is lying on a smooth floor. A force $\to F$ is applied at the point P parallel to AB, such that the object has only the translational motion without rotation.
To find the location of P with respect to C.
Solution:
To have the required motion, the force $\to F$ has to be applied at the center of mass.
The point P must be the centre of mass of the T-shaped object since the force F does not produce any rotational motion of the object.
The horizontal part of the T-shaped object has length L. If the mass of the horizontal portion is ‘m’, the mass of the vertical portion of the T- shaped object is 2m since its length is 2L.
Therefore, the T-shaped object reduces to two point masses m and 2m at distances 2L and L respectively from the point C. The distance ‘r’ of the centre of mass of the system from the point C is given by
$r = \frac{m_{1} r_{1} + m_{2} r_{2}}{m_{1} + m_{2}} ⟹ r = \frac{m \times 2 L + 2 m \times L}{m + 2 m} ⟹ r = \frac{4 L}{3}$
is the location of P with respect to C.

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