Question

four thin uniform rods of mass m and length are joined to form a square. what will be the moment of inertia about an axis along its one diagonal?

Answer 1

Answer on Question#39651, Physics, Mechanics | Kinematics | Dynamics

Question:

Four thin uniform rods of mass $m$ and length are joined to form a square. What will be the moment of inertia about an axis along its one diagonal?

Answer:

Moment of inertia for one rod equals:

$$I _ {1} = \int d m r ^ {2}$$ $$d m = \frac {m}{l} d x$$ $$r = \frac {x}{\sqrt {2}}$$

Therefore:

$$I _ {1} = \int_ {0} ^ {l} \frac {m}{l} d x \left(\frac {x}{\sqrt {2}}\right) ^ {2} = \frac {m l ^ {2}}{6}$$

Therefore, total moment of inertia equals:

$$I = 4 I _ {1} = \frac {2}{3} m l ^ {2}$$

Answer: $\frac{2}{3}ml^2$