Two discs are fastened together, so they act as one object with a common axis of

Question

Two discs are fastened together, so they act as one object with a common axis of rotation. Radius 1 = 3 m and Radius 2 = 4.5 m. Each disc is constructed of a material with a density of 20kg/m^2. Two masses hang from thin mass-less cables. Refer to picture.

a) Calculate the moment of inertia of the composite disc.

b) If Mass 1= 66 kg, what is Mass 2 (assume no angular acceleration in the system)?

c) If 33 kg is added to Mass 1 ,and Mass 2 is the same as part B, calculate the angular acceleration of the discs, the linear accelerations of both masses and the tensions in both cables.

THANKS!

Explanation / Answer

a) I = 0.5(m1*r1^2 + m2*r2^2)

= 0.5*(p*pi*r1^2*r1^2 + p*pi*r2^2*r2^2)

= 0.5*p*pi(r1^4 + r2^4)

= 491.1 kgm^2

b) 491.1 = 0.5*(66*3^2 + m2*4.5^2)

=> m2 = 19.17 kg

c) Inew = 0.5(99*3^2 + 19.17*4.5^2)

= 639.6 kgm^2

m1*g - T1 = m1*a1

T2 - m2*g = m2*a2

T1*r1 - T2*r2 = (Inew*alpha)

alpha = a1/r1 = a2/r2

m1*r1*(g - r1*alpha) - m2*r2*(g+r2*alpha) = Inew*alpha

=> alpha = (m1r1 - m2r2)g / (Inew + m1r1^2 + m2r2^2)

=>alpha = 1.098 rad/s^2

a1 = 3.29 m/s^2

a2 = 4.94 m/s^2

T1 = 664.29 N

T2 = 286.4 N

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