Two objects are attached to ropes that are attached to wheels on a common axle a

Question

Two objects are attached to ropes that are attached to wheels on a common axle as shown below. The two wheels are glued together so that they form a single object. The total moment of inertia of the two wheels is 45 kg · m2. The radii of the wheels are R1 = 1.1 m and R2 = 0.10 m.

(a) If m1 = 24 kg, find m2 such that there is no angular acceleration of the wheels.
  kg

(b) If 12 kg is gently added to the top of m1, find the angular acceleration of the wheels.
  rad/s2
Find the tension in the rope holding m1.
kN
Find the tension in the rope holding m2.
kN

Explanation / Answer

the mass m2 is calculated as follows:

m2 = [R1/R2]m1 = [1.1/0.1] *24 = 264 kg

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the angular acceleration is calculated as follows:

alpha = [m1R1 - m2R2] / m1R12 + m2R22 + I0

=  [36*1.1 - 264*0.1] /[36*1.1*1.1 +264*0.1*0.1 + 45 ]

= 0.145 rad/s2

the tension T1 is calculated as follows;

T1 = m1[g - R1*alpha] = 36*[9.81 - 1.1*0.145] = 347.418 N = 0.347 kN

The tension T2 is calculated as follows:

T2 =  m2[g + R2*alpha] = 36*[9.81 + 0.1*0.145] = 353.682 N = 0.354 kN

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