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Question

A uniform T shaped object with dimensions as shown in the figure, is lying on a smooth floor. A force F is applied at the point P parallel to limb AB, such that the object has only the translational motion without rotation. Find the distance of point P from C


A
2l3
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B
3l2
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C
4l3
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D
l
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Solution

The correct option is C 4l3
F must be directed towards the centre of mass of system, so that it passes through COM and τext=0
If τext=0 then angular acceleration of body α=0, which means that rotational aspect of motion will be absent, and body will be in pure translational motion.
T shape is symmetrical about y axis hence, COM will lie along y axis only.

Considering origin O at the centre of horizontal limb of length l and axis as shown in figure:


since (length mass), for uniform rod
m1=m, m2=2m
m2vertical limb of length 2l

Replacing the individual limbs with a point mass placed at their respective COM:
y1=0, y2=l
yCM=m1y1+m2y2m1+m2
yCM=(m×0)+(2m×l)m+2m=2ml3m
yCM=2l3
PC=COPO=(2l2l3)
Distance of point P with respect to C,
PC=4l3

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