Chapter 9 Problem 77

Problem

Two people, one of mass 85 kg and the other of mass 55 kg, sit in a rowboat of mass 78 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat, 3.0 m apart from each other, now exchange seats. How far and in what direction will the boat move?

Solution

Set the origin at 55 kg person.

$x_{\text{CM}}={\frac {m_{A}x_{A}+m_{B}x_{B}+m_{C}x_{C}}{m_{A}+m_{B}+m_{C}}}$

$={\frac {\left(55{\text{kg}}\right)\left(0\right)+\left(78{\text{kg}}\right)\left(1.5{\text{m}}\right)+\left(85{\text{kg}}\right)\left(3.0{\text{m}}\right)}{218{\text{kg}}}}$ $=1.706{\text{m}}$

when they exchange positions, the boat moves by d, but the location of the center of mass remains the same, since there are no external forces.

$x_{\text{CM}}={\frac {m_{A}x_{A}+m_{B}x_{B}+m_{C}x_{C}}{m_{A}+m_{B}+m_{C}}}$

$1.706{\text{m}}={\frac {\left(85{\text{kg}}\right)\left(d\right)+\left(78{\text{kg}}\right)\left(d+1.5{\text{m}}\right)+\left(55{\text{kg}}\right)\left(d+3.0{\text{m}}\right)}{218{\text{kg}}}}$ $={\frac {\left(218d{\text{kg.m}}\right)+\left(282{\text{kg.m}}\right)}{218{\text{kg}}}}$

$d=0.412{\text{m}}$