Question

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

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

Step-by-step solution

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.

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).

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.

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.

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 \(m_{Judy} = 59.0 \mathrm{~kg}\).

Key concepts

Conservation of Momentum

Conservation of momentum is a fundamental principle in physics. It states that in a closed system, the total momentum remains constant if no external forces act on it.
This concept helps us understand various scenarios, like two people changing seats in a canoe. Here, the key point is that although the canoe itself might move, the overall system's momentum does not change. Imagine Richard and Judy switching seats. Their movement causes the canoe to shift, but the lack of external forces means the center of mass of the entire system (including the canoe) doesn't.
This principle ensures that, despite the canoe's visible motion relative to an external object like a log, the internal configuration's momentum stays balanced. This equilibrium allows us to predict other values, such as someone's unknown mass, by analyzing positional changes within the system.

Mass Calculation

Mass calculation in physics requires using various known parameters to determine unknown values. For this exercise, the goal was to find Judy's mass by considering how the system's center of mass stayed the same despite individual movements.
To start, understand the components involved:

• The canoe's mass: 31.6 kg
• Richard's mass: 78.4 kg
• Judy's mass: Unknown ( \( m_{Judy} \) )
• The distance moved by the canoe: 0.412 meters relative to the initial setup

Use the formula for the center of mass: \[ x_{cm,i} = \frac{m_{canoe} x_{canoe} + m_{Richard} x_{Richard} + m_{Judy} x_{Judy}}{M} \]After determining the final and initial center of mass using their respective positions before and after the movement, the equations will allow us to solve for the unknown mass. By setting initial and final calculations equal to each other, we rearrange to isolate Judy’s mass. Ultimately, this approach lets us see that Judy's mass is 59.0 kg.

Symmetric Motion

Symmetric motion plays a crucial role in understanding the dynamics of the problem. Initially, Richard and Judy were seated symmetrically with respect to the canoe's center, allowing us to use symmetry to simplify calculations.
Symmetry implies that before switching, their respective positions exert equal and opposite effects on the system's balance. This arrangement helps ensure that when they swap positions, any changes to the system can be directly attributed to their movement patterns rather than external forces.
The problem describes their motion as symmetric, with Richard moving to where Judy was and vice versa. During this symmetry, the canoe moved relative to an external point (the submerged log), but internally, symmetry kept the center of mass constant. This consistent center of mass is the pivot for finding unknown values, such as Judy's mass. By understanding symmetric motion, we realize how changes in position cancel each other out, maintaining the system's overall stability.