Question

Two students sit on either side of a teeter-totter that is \(2.8 \mathrm{~m}\) in length. The teeter-totter balances when the student on the left side is \(1.1 \mathrm{~m}\) from the center and the student on the right is \(1.4 \mathrm{~m}\) from the center. The total mass of the two students is \(84 \mathrm{~kg}\). What is the mass of the student on the left side of the teeter- totter? (Assume that the teeter-totter itself pivots at the center and produces zero torque.)

Short answer

The mass of the student on the left side is approximately 47 kg.

Step-by-step solution

Understanding the Problem

We need to find the mass of the student on the left side of the teeter-totter given that it balances. We know the distances from the center for both students and their combined mass.

Applying the Principle of Torque Balance

For the teeter-totter to balance, the clockwise torque should equal the counterclockwise torque. Torque is calculated as the product of force (weight in this case) and distance from the pivot. Mathematically, this is given by:\[ m_1 \cdot d_1 = m_2 \cdot d_2 \]where \(m_1\) is the mass of the student on the left, \(d_1 = 1.1\, \mathrm{m}\), \(m_2\) is the mass of the student on the right, and \(d_2 = 1.4\, \mathrm{m}\).

Setting Up the Equation with Known Values

Substitute the known values into the torque balance equation:\[ m_1 \cdot 1.1 = m_2 \cdot 1.4 \]

Expressing One Mass in Terms of the Other

We know the total mass of the two students is 84 kg, so we can write:\[ m_1 + m_2 = 84 \]Solve the two equations together:1. \( m_1 \cdot 1.1 = m_2 \cdot 1.4 \)2. \( m_1 + m_2 = 84 \)From equation 2, express \( m_2 \) as \( m_2 = 84 - m_1 \).

Solving the System of Equations

Substitute \( m_2 = 84 - m_1 \) into the torque equation:\[ m_1 \cdot 1.1 = (84 - m_1) \cdot 1.4 \]Distribute and simplify:\[ 1.1m_1 = 117.6 - 1.4m_1 \]Bring like terms together:\[ 1.1m_1 + 1.4m_1 = 117.6 \]\[ 2.5m_1 = 117.6 \]

Calculating the Mass of the Left Student

Solve for \( m_1 \) by dividing both sides by 2.5:\[ m_1 = \frac{117.6}{2.5} \approx 47.04 \]

Conclusion

The mass of the student on the left side of the teeter-totter is approximately 47 kg.

Key concepts

Teeter-totter

A teeter-totter, often known as a seesaw, is a classic playground apparatus that provides an excellent real-world example of basic physics concepts such as torque and balance. It consists of a long, flat board that is balanced on a central pivot point. When two individuals sit on opposite ends, the board moves up or down based on the forces they exert at their respective distances from the center.

This concept is crucial in understanding how differing weights at varying distances can achieve equilibrium, or balance. When a teeter-totter is balanced, the rotational forces, known as torques, on either side of the pivot are equal. This allows us to apply the principles of physics to determine unknowns, such as someone's weight in this case.

Balance

Balance is all about achieving torque equilibrium. In the context of a teeter-totter, balance is achieved when the rotational forces exerted by weights on either side are equal. For instance, if two people on a teeter-totter are equidistant from the center and weigh the same, the seesaw will perfectly keep level.

However, if one person is heavier or sits further from the pivot, the system relies on varying the distance to maintain balance. This makes the concept of balance essential in problems involving torques, where objects must be positioned in a way that ensures the net torque equals zero. Hence, teeter-totters are a fun way of experiencing the physics of balance in real life.

Torque Equation

Torque is a measure of how much a force causes an object to rotate around an axis. It is calculated using the formula: \[ \tau = F imes r \] where \( \tau \) is the torque, \( F \) is the force applied (weight, in the context of gravity problems), and \( r \) is the distance from the pivot point.

In the setting of a teeter-totter, we typically deal with the force due to gravity, expressed as weight, which is mass times the acceleration due to gravity \( g \). For balanced systems, the torques produced on either side are equal, described by this equation:\[ m_1 \cdot d_1 = m_2 \cdot d_2 \] This means the product of mass and distance for one side matches that of the other side, allowing us to solve for unknown masses or positions.

Mass Calculation

Calculating mass in this context involves solving equations derived from torque balance. For the teeter-totter problem, you perform the following calculations to find the mass of the student on one side:1. Determine the relationship from the balance condition, \( m_1 \cdot 1.1 = m_2 \cdot 1.4 \).2. Use the total given mass, \( m_1 + m_2 = 84 \) kg, to express one mass in terms of the other.3. Substitute \( m_2 = 84 - m_1 \) into the first equation.4. Simplify and solve: \[ m_1 \cdot 1.1 = (84 - m_1) \cdot 1.4 \]5. Rearrange to find \( m_1 \): \[ 2.5m_1 = 117.6 \] 6. Solve by dividing: \[ m_1 = \frac{117.6}{2.5} \approx 47 \] This calculation shows how the principles of torque and balance help solve for a specific mass, using given data.