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 39 kg · m2. The radii of the wheels are R1 = 1.5 m and R2 = 0.50 m.

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

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

Explanation / Answer

a)
Since there is no angular acceleration, a=0
Hence net torque will be 0
so, torque due to m1 must balance torque due to m2
m1*g*R1= m2*g*R2
m1*R1= m2*R2
24*1.5 = m2*0.5
m2= 72 Kg

b)
Net torque = (m1+m)*g*R1 - m2*g*R2
= (24+12)*9.8*1.5 - 72*9.8*0.5
=176.4 nm

net torque = I*a
176.4 = 39*a
a= 4.52 rad/s^2

3)

linear acceleration, a' = R1*a = 1.5*4.52=6.78 m/s^2
T= m1*(g-a')
= 24*(9.8-6.78)
=72.48 N

4)

linear acceleration, a' = R2*a = 0.5*4.52=2.26 m/s^2
T= m2*(g+a')
= 72*(9.8+2.26)
=868.32 N

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