Where should Kariann and Aysha place the pivot point of the seesaw? Assume that the seesaw is massless.
Understand the Problem
The question is asking for the location of the pivot point on a seesaw balancing two individuals of different weights, requiring the calculation of torques and distances to find the equilibrium point.
Answer
The pivot point should be placed approximately $0.99 \text{ m}$ from Aysha.
Answer for screen readers
The pivot point should be placed approximately $0.99 \text{ m}$ from Aysha and $0.76 \text{ m}$ from Kariann.
Steps to Solve
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Define known values The masses of Kariann ($m_k = 56 \text{ kg}$) and Aysha ($m_a = 43 \text{ kg}$) contribute to the forces they exert due to gravity.
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Kariann's weight: $$ F_k = m_k \cdot g = 56 \text{ kg} \times 9.8 \text{ N/kg} = 549 \text{ N} $$
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Aysha's weight: $$ F_a = m_a \cdot g = 43 \text{ kg} \times 9.8 \text{ N/kg} = 421.4 \text{ N} $$
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Define the distances Let $r_a$ be Aysha's distance from the pivot. Therefore, Kariann's distance from the pivot can be expressed as: $$ r_k = 1.75 \text{ m} - r_a $$
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Set up the torque equilibrium equation For the seesaw to be balanced, the torques around the pivot must be equal: $$ F_k \cdot r_k = F_a \cdot r_a $$
Substituting the expressions for the forces and distances: $$ 549 \cdot (1.75 - r_a) = 421.4 \cdot r_a $$
- Solve for $r_a$ Expanding and rearranging the equation: $$ 549 \times 1.75 - 549 r_a = 421.4 r_a $$
Combine the $r_a$ terms: $$ 549 \times 1.75 = 421.4 r_a + 549 r_a $$ $$ 549 \times 1.75 = (421.4 + 549) r_a $$ $$ r_a = \frac{549 \times 1.75}{421.4 + 549} $$
- Calculate the distance Now substituting into the equation: $$ r_a = \frac{961.75}{970.4} \approx 0.99 \text{ m} $$
Kariann's distance: $$ r_k = 1.75 - r_a \approx 1.75 - 0.99 \approx 0.76 \text{ m} $$
The pivot point should be placed approximately $0.99 \text{ m}$ from Aysha and $0.76 \text{ m}$ from Kariann.
More Information
The calculation shows that Aysha sits further from the pivot, which balances the heavier weight of Kariann, demonstrating practical applications of torque and equilibrium in physics.
Tips
- Forgetting to convert masses to weights by including the acceleration due to gravity.
- Mixing up the distances or not keeping track of which distance corresponds to which person.
- Neglecting to correctly sum torques from both sides of the pivot.
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