- #36

Steve4Physics

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- 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##.