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Systems of Particles and Rotational Motion

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Question : 14 of 33
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A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping.
Solution:  
Mass of hollow cylinder,M=3 kgM = 3\,\mathrm{kg}
Radius of hollow cylinder, R=40 cm=0.4 mR = 40\,\mathrm{cm} = 0.4\,\mathrm{m}
M.I. of the hollow cylinder about its axis
I=MR2=3 kg×(0.4 m)2=0.48 kg m2I = MR^2 = 3\,\mathrm{kg} \times (0.4\,\mathrm{m})^2 = 0.48\,\mathrm{kg}\,\mathrm{m}^2
Force F=30 NF = 30\,\mathrm{N}
∴ Torque, τ=F×R=30 N×0.4 m\tau = F \times R = 30\,\mathrm{N} \times 0.4\,\mathrm{m} =12 N m= 12\,\mathrm{N}\,\mathrm{m}
Also τ=Iα\tau = I \alpha
α=τI=120.48=25 rad s−2\alpha = \frac{\tau}{I} = \frac{12}{0.48} = 25\,\mathrm{rad}\,\mathrm{s}^{-2}
Linear acceleration, a=Rα=0.4×25=10 m s−2a = R \alpha = 0.4 \times 25 = 10\,\mathrm{m}\,\mathrm{s}^{-2}
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