problem set 6 solutions问题集解决方案.pdfVIP

Physics200

ProblemSet6

Solution

1.(i)WhatisthemomentofinertiaICMofapropellerwiththreeblades(treatedasrods)ofmass

m,lengthL,at120◦relativetoeachother?(ii)Ifatorqueτactsonthishowlongwillittaketo

reachanangularvelocityω?(iii)Howmanyrevolutionswillithavemadebeforereachingthisω?

(iv)GetthenumericalanswersifL=1.25m,m=12kg,τ=3000N·m,ω=2000rad/s.

Answer:

(i)Weknow(ifnot,lookonp.296)thatthemomentofinertiaofasinglerodrotatingaround

itsendis1mL2.It’snothardtoconvinceoneselfthatiftherearethreeofthemrotating

3

aroundthesameaxisandinthesameplane,themomentofinertiaisjustthreetimesthis,

ICM=mL2.

(ii)Sinceω=αtandτ=ICMα,

ICMωmL2ω

t==.

ττ

(iii)Fromourknowledgeofconstantaccelerationproblems,

2ω2ICMω2mL2ω2

ω=2αθ=⇒θ===.

2α2τ2τ

Thenumberofrevolutionsitmadeis

θmL2ω2

N==.

2π4πτ

(iv)

ICM=(12kg)(1.25m)2=19kg·m2,

(12kg)(1.25m)2(2000s−1)

t==13s,

2(3000