Question

A pendulum is constructed from a heavy metal rod and a metal disk, both of uniform mass density. The center of the disk is bolted to one end of the rod, and the pendulum hangs from the other end of the rod. The rod has a mass of =1.0 kg and a length of =49.8 cm. The disk has a mass of =4.0 kg and a radius of =24.9 cm. The acceleration due to gravity is =9.8 m/s2.
The pendulum is held with the rod horizontal and then released. What is the magnitude of its angular acceleration at the moment of release?

Answer 1

Answer: α = 20.4 rad/s²

Explanation:

α = τ/I

If we model it as a rod and a point mass, the moment of inertia

I = mL²/3 + ML² = 0.498²)(1.0/3 + 4.0) = 1.07 kg•m²

torque will be

τ = 1.0(9.8)(0.498/2) + 4.0(9.8)(0.498) = 21.96 N•m

α = 21.96 / 1.07 = 20.43...

Another possibility is that we model the large mass as a disc and apply the parallel axis theorem

I = mL²/3 + ½MR² + ML²

I =  1.07 + ½(4.0)0.249² = 1.19 kg•m/s

then

α = 18.45... 18.5 rad/s²