What is the average power dissipated by a 2kg grinding wheel of radius 0.1 m brought to rest in 10 rev from an initial velocity of 3000 rpm? Assume constant acceleration.
n=10n=10n=10
ω=3000 rpm=314 rad/s\omega=3000\ rpm=314\ rad/sω=3000 rpm=314 rad/s
I=mr2/2=2⋅0.12/2=0.01 kg⋅m2I=mr^2/2=2\cdot0.1^2/2=0.01\ kg\cdot m^2I=mr2/2=2⋅0.12/2=0.01 kg⋅m2
W=ΔKE=Iω22=0.01⋅31422≈493 (J)W=\Delta KE=\frac{I\omega^2}{2}=\frac{0.01\cdot314^2}{2}\approx493\ (J)W=ΔKE=2Iω2=20.01⋅3142≈493 (J)
P=W/tP=W/tP=W/t
ϕ=ωt−ϵt2/2\phi=\omega t-\epsilon t^2/2ϕ=ωt−ϵt2/2
ωf=ω−ϵt→ϵ=(ω−ωf)/t=(ω−0)/t=ω/t\omega_f=\omega-\epsilon t\to \epsilon=(\omega-\omega_f)/t=(\omega-0)/t=\omega/tωf=ω−ϵt→ϵ=(ω−ωf)/t=(ω−0)/t=ω/t
ϕ=ωt−ϵt2/2=ωt−ωt22t=ωt2\phi=\omega t-\epsilon t^2/2=\omega t-\frac{\omega t^2}{2t}=\frac{\omega t}{2}ϕ=ωt−ϵt2/2=ωt−2tωt2=2ωt
2πn=ωt2→t=4πnω=4⋅3.14⋅10314=0.4 (s)2\pi n=\frac{\omega t}{2}\to t=\frac{4\pi n}{\omega}=\frac{4\cdot 3.14\cdot10}{314}=0.4\ (s)2πn=2ωt→t=ω4πn=3144⋅3.14⋅10=0.4 (s)
P=W/t=493/0.4=1232.5 (W)P=W/t=493/0.4=1232.5\ (W)P=W/t=493/0.4=1232.5 (W) . Answer
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