Answer to Question #118177 in Mechanics | Relativity for rishal deo

Question #118177
A flywheel has a moment of inertia of 1.6 × 10−3 kg.m2. When a constant torque is applied, it reaches an angular velocity of 1200 rev/min in 15 s. Assuming it started from rest, find:
i) the angular acceleration
ii) the unbalanced torque applied
iii) the angle turned through 15 s
iv) the work done on the flywheel by the torque
1
Expert's answer
2020-05-26T12:49:05-0400

Given, Moment of inertia I=1.6×103kg.m2I=1.6 × 10^{−3} kg.m^2 ,

Initial angular speed ω0=0\omega_0=0 ,final angular speed ω=1200×2π60=40πrad/s\omega=1200\times \frac{2\pi}{60}=40\pi \:rad/s ,time t=15st=15s ,

i). Angular acceleration

α=ωω0t    α=40π15=8π3rad/s2\alpha=\frac{\omega-\omega_0}{t}\implies \alpha=\frac{40\pi}{15}=\frac{8\pi}{3} \: rad/s^2

ii).

Unbalanced torque

τ=Iα    τ=1.6×103×8π3=13.40×103Nm\tau=I\alpha\\ \implies \tau=1.6\times 10^{-3}\times \frac{8\pi}{3}=13.40\times 10^{-3}\:Nm

iii). Let the required angle is θ\theta , thus

ω2=ω02+2αθ    θ=ω2ω022α    θ=(40π)228π3=942.47rad\omega^2=\omega_0^2+2\alpha \theta\\ \implies \theta=\frac{\omega^2-\omega_0^2}{2\alpha}\\ \implies \theta=\frac{(40\pi)^2}{2\frac{8\pi}{3}}=942.47 \: rad

Alternatively, we can apply

θ=ω0t+12αt2\theta=\omega_0t+\frac{1}{2}\alpha t^2

and eventually we will get the same result.



iv).

By applying work energy theorem for rotational motion we get,


Wrot=12Iω212Iω02W_{rot}=\frac{1}{2}I\omega^2-\frac{1}{2}I\omega_0^2

Thus we get,

Wrot=12×1.6×103(40π)2=12.63JW_{rot}=\frac{1}{2}\times 1.6\times 10^{-3} (40\pi)^2=12.63J

Hence we done.


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