Question

An equilateral triangular wire frame is made from 3 rods of equal mass and length \( \ell \) each. The frame is rotated about an axis perpendicular to the plane of the frame and passing through its end. What is the radius of gyration of the frame? (a) \( \frac{\ell}{2} \) (b) \( \ell \) (c) \( \frac{\ell}{\sqrt{2}} \) (d) \( \frac{\ell}{2 \sqrt{3}} \)

Answer 1

Correct option is (c) \(\frac{l}{\sqrt{2}}\)

The MoI of a Rod of mass M and length L that passes through the center of the rod perpendicular to the rod itself will be = 1/12 ml²

The MoI of a Rod of mass M and length L that passes through one end of the rod perpendicular to the rod itself will be = 1/3 ml²

AD = √3/2L

MoI of BC about the axis through perpendicular to the plane of the triangle will be -

= 1/12ml² + m(√3/2l)²

= 1/12ml² + 3/4ml²

= 5/6ml²

According to the principle of superposition -

= 1/3ml² + 1/3ml² + 5/6ml²

= 9/6ml²

= 3/2ml²

Therefore, the radius will be -

k = √3/2ml²/3m

= l/√2