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NEET PHYSICSMedium

Four identical thin rods each of mass MM and length tt, form a square frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is:

A

43Mt2\frac{4}{3}Mt^2

B

23Mt2\frac{2}{3}Mt^2

C

133Mt2\frac{13}{3}Mt^2

D

13Mt2\frac{1}{3}Mt^2

Step-by-Step Solution

Let us consider one of the four rods. The moment of inertia of a thin rod of mass MM and length tt about an axis passing through its own centre of mass and perpendicular to its length is ICM=Mt212I_{CM} = \frac{Mt^2}{12}. The perpendicular distance from the centre of the square to the centre of each rod is d=t2d = \frac{t}{2}. According to the parallel axis theorem, the moment of inertia of one rod about the axis passing through the centre of the square and perpendicular to its plane is: I1=ICM+Md2=Mt212+M(t2)2=Mt212+Mt24=4Mt212=Mt23I_1 = I_{CM} + Md^2 = \frac{Mt^2}{12} + M\left(\frac{t}{2}\right)^2 = \frac{Mt^2}{12} + \frac{Mt^2}{4} = \frac{4Mt^2}{12} = \frac{Mt^2}{3} Since the square frame consists of 4 identical rods, the total moment of inertia of the frame is: Itotal=4×I1=4×Mt23=43Mt2I_{total} = 4 \times I_1 = 4 \times \frac{Mt^2}{3} = \frac{4}{3}Mt^2

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