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Chapter 7: Problem 8

Richard, mass \(78.4 \mathrm{~kg}\), and Judy, who is less massive, are enjoying Lake George at dusk in a \(31.6\) -kg canoe. When the canoe is at rest in the placid water, they change seats, which are \(2.93 \mathrm{~m}\) apart and symmetrically located with respect to the canoe's center. Richard notices that the canoe moved \(41.2 \mathrm{~cm}\) relative to a submerged log and calculates Judy's mass. What is it?

Short Answer

Expert verified
Judy's mass is \(59.0 \mathrm{~kg}\).

Step by step solution

01

Analyze the Situation

Let \(x_{cm,i}\) be the initial center of mass of the system (canoe + people) and \(x_{cm,f}\) the final one. We know that the center of mass remains at the same place because we assumed there were no outside forces. So, even though the canoe moves, the system including the persons doesn’t. This tells us that \(x_{cm,i} = x_{cm,f}\). We calculate these two quantities.
02

Calculate the Initial Center of Mass

The initial center of mass includes the canoe and the two paddlers, Richard and Judy. We can calculate this by using the formula for the center of mass of a system:\[x_{cm,i} = (m_{canoe} x_{canoe} + m_{Richard} x_{Richard} + m_{Judy} x_{Judy}) / M\]where M is the total mass of the system. Positions \(x_{canoe}\), \(x_{Richard}\), and \(x_{Judy}\) are, respectively, 0 (since we take the center of the canoe as reference), -1.465m, and +1.465m (since Richard and Judy are initially symmetrically located with respect to the canoe's center).
03

Calculate the Final Center of Mass

After Richard and Judy have changed their places, the final center of mass will be calculated similarly to Step 2. Now the positions have changed, \(x_{Richard}\) is now +1.465m and \(x_{Judy}\) now -1.465m, but the canoe has moved \(41.2 \mathrm{~cm}\) = 0.412m, thus \(x_{canoe}\) is now -0.412m.
04

Equate the Expressions

Since the center of mass remains the same, the expression found in Step 2 (initial center of mass) equals the expression found in Step 3 (final center of mass). Equating them, we can solve for Judy's mass \(m_{Judy}\), as all other variables are known.
05

Solve for Judy's Mass

After isolating \(m_{Judy}\) on one side of the equation, we get an expression for Judy's mass in terms of known quantities. After calculating these values, her mass is determined to be \(m_{Judy} = 59.0 \mathrm{~kg}\).

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