find the tension in the threada thread is wound

Question

Find the tension in the thread:

A thread is wound round a heavy homogeneous cylinder of mass m and radius. The cylinder is permitted to fall from rest and unwinds the thread. Find the tension in the thread and angular acceleration of the cylinder.

Solution

Free body diagram of the cylinder as it starts rolling down. Its angular acceleration α and angular velocity ω are supposed to be in the direction shown. The origin of the fixed axes x, y is taken as the point of support of the thread and the moving z axis is taken as passing through C away from the front side.

Three unknowns α, a_cy  = d ² y/ dt ²  and the tension T in the thread ought to be determined in this problem. The unknowns α and acy are not independent but can be related to each other. The straight length OA of the string does not move. The velocity of point A must be zero. Then considering C as pole, we have

V_A = V_C + V_CA = 0

∴ Vc y - a. ω = 0

Differentiating the above expression, we get

 a _{c y} = a . dω/ dt = aα               --------- (1)

Now, we may write dynamic equilibrium equations for the cylinder

∑ F_y  = 0,        T - mg + ma_cy  = 0          --------- (2)

∑ M _C = 0, T. a - I _C α = 0

We have

I_C = ma ²/2

∴ We get

T. a = (ma ²/2). α

∴ T = (ma/2) α          ------ (3)

 And from Eq. (2), we obtain

T = m_g - ma_cy

= mg - ma α - =( ½) ma α

∴ (3/2) ma α = mg

α = 2 g/3 a

Substituting this value of α in Eq. (3), we obtain

T = ma α = (ma/ 2) (2/3) (g/a)

= (1/3) mg

And

a_cy  =(2/3)g