Determine the angular acceleration of the motion:

A flywheel of mass 15 kg and radius 20 cm is wound by a rope that carries a weight A of mass 5 kg at its free end as illustrated in Figure. Determine the angular acceleration of the motion, supposing gravitational acceleration = 10 m/sec² for purpose of simplified calculations, and letting the following two cases:

(i) If friction at the bearing of flywheel-shaft is zero.

(ii) If frictional couple developed at the bearing of shaft is 2 N-m.

Solution

While block A travels a distance s downward, this has certain linear velocity V and acceleration a. The equivalent angular velocity ω and angular acceleration α of the flywheel of radius (r = 0.2 m) are specified by

V = r ω = 0.2 ω

a = r α = 0.2 α

s = r θ

Here, θ is the angular displacement of the flywheel. Likewise to equation in linear motion given by

                                            V ² = 2 a s

The equation in angular velocity is

                                           ω² = 2 α θ

By using principle of conservation of energy letting datum level as A′ which is the final position of A, we have

W_A   × s = W V ²  /2g + I_m  ω²  /2+ C × θ

 (i)        While frictional couple C = 0;

5 × 10 × (0.2 θ) = 5 (0.2)²/2  ω² + 15 (0.2)²  /2 × ω²/2  + 0

10 ×θ = ω² × (0.2) ² /2[5 + 7.5]

10 ×θ = α θ × (0.2)² × 12.5

∴ α = 20 rad / sec ^2.

 (ii)  50 × 0.2 θ = α θ (0.2)² × 12.5 + 2 × θ

∴ α = 16 rad / sec^2.