- #36
Steve4Physics
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It may not be the original intention of the question, but assume that the question
- specifies the river’s width (or more correctly the distance between pick-up and drop-off points) as ##W##;
- specifies lateral drift of the raft is negligible;
- specifies that ##k## is variable;
- requires us to find the minimum value of the cylinder’s angular displacement (##\alpha##) resulting from varying ##k##. I.e. we need to minimise ##\alpha (k)##.
##\alpha ##= angular displacement of cylinder (mass M) relative to the ground.
##\beta## = angle swept by rod relative to ground.
Conserving angular momentum gives:
##\frac 12 Mr^2 α + m(kr)^2 β = 0##
##α = -\frac {2mk^2} M β##
The geometry gives:
##\sin ({\frac β2}) = \frac {W/2}{kr} = \frac W {2kr}##
##β = 2\sin^{-1} (\frac W {2kr})##
We can adopt a sign-convention which makes ##β## negative (in order to make ##\alpha## positive, for neatness); so use:
##β = -2\sin^{-1} (\frac W {2kr})##
Combining the equations for ##α## and ##β##:
##α = \frac {4mk^2} M \sin^{-1} (\frac W {2kr})##
Anyone so inlined can then determine the minimum value of α(k) by starting with ##\frac {dα}{dk} = 0##.
- specifies the river’s width (or more correctly the distance between pick-up and drop-off points) as ##W##;
- specifies lateral drift of the raft is negligible;
- specifies that ##k## is variable;
- requires us to find the minimum value of the cylinder’s angular displacement (##\alpha##) resulting from varying ##k##. I.e. we need to minimise ##\alpha (k)##.
##\alpha ##= angular displacement of cylinder (mass M) relative to the ground.
##\beta## = angle swept by rod relative to ground.
Conserving angular momentum gives:
##\frac 12 Mr^2 α + m(kr)^2 β = 0##
##α = -\frac {2mk^2} M β##
The geometry gives:
##\sin ({\frac β2}) = \frac {W/2}{kr} = \frac W {2kr}##
##β = 2\sin^{-1} (\frac W {2kr})##
We can adopt a sign-convention which makes ##β## negative (in order to make ##\alpha## positive, for neatness); so use:
##β = -2\sin^{-1} (\frac W {2kr})##
Combining the equations for ##α## and ##β##:
##α = \frac {4mk^2} M \sin^{-1} (\frac W {2kr})##
Anyone so inlined can then determine the minimum value of α(k) by starting with ##\frac {dα}{dk} = 0##.