Four particles, each of mass 0.11 kg, are placed at the vertices of a square wit

Question

Four particles, each of mass 0.11 kg, are placed at the vertices of a square with sides of length 0.41 m. The particles are connected by rods of negligible mass. This rigid body can rotate in a vertical plane about a horizontal axis A that passes through one of the particles. The body is released from rest with rod AB horizontal, as shown in the figure. (a) What is the rotational inertia of the body about axis A? (b) What is the angular speed of the body about axis A at the instant rod AB swings through the vertical position?

Explanation / Answer

a) moment of inertia of point mass about axis at a distance of r is given by mr^2 .

if a systen contains many point mass then moment of inertia of system about an axis given by

I = I1 + I2 + I3 + .. ... = m1 r1^2 + m2 r2^2 + m3 r3^3 + .....

I = (0.11 x 0.41^2 ) + (0.11 x 0.41^2 ) + (0.11 x (sqrt(0.41^2 + 0.41^2)^2)

I = 0.074 kg m^2

b) as rod AB goes from horizontal to vertical,

centre of mass lowers it position by 0.41/2 = 0.205 m

using energy conservation,

mgh = Iw^2 /2

(4 x 0.11) x 9.81 x (0.41/2) = 0.074 w^2 /2

w = 4.89 rad/s

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