Question

A 25 -kg child sits on one side of a teeter-totter, at a distance of \(2 \mathrm{~m}\) from the pivot point. A mass \(m\) is placed at a distance \(d\) on the other side of the pivot, in an effort to balance the teeter-totter. Which of the following combinations of mass and distance (A, B, C, or D) balances the teeter-totter? (Assume that the teeter-totter itself pivots at the center and produces zero torque.) $$ \begin{array}{|l|l|l|l|l|} \hline & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline \text { Mass, } \boldsymbol{m} & 10 \mathrm{~kg} & 50 \mathrm{~kg} & 40 \mathrm{~kg} & 20 \mathrm{~kg} \\ \hline \text { Distance, } \boldsymbol{d} & 2 \mathrm{~m} & 1 \mathrm{~m} & 1.5 \mathrm{~m} & 2.5 \mathrm{~m} \\ \hline \end{array} $$

Short answer

Options B and D balance the teeter-totter.

Step-by-step solution

Understanding the Concept

In this exercise, we need to balance a seesaw (teeter-totter) by using the concept of torque. Torque is the product of force and the distance from the pivot point (lever arm). The seesaw is balanced when the torque on one side is equal to the torque on the other side.

Write the Torque Equation

The torque produced by the child is given by \( \tau_1 = F_1 \times d_1 \), where \( F_1 \) is the force due to the child's weight and \( d_1 \) is the distance from the pivot. The force is calculated as weight, \( F_1 = m_1 \times g \), where \( m_1 = 25 \text{ kg} \) and \( g = 9.8 \text{ m/s}^2 \). So, \( \tau_1 = 25 \times 9.8 \times 2 \).

Calculate Torque from Child's Side

Calculating the torque from the child's side: \[ \tau_1 = 25 \times 9.8 \times 2 = 490 \text{ Nm} \]

Write Torque Equation for Mass \( m \)

The torque produced by the mass \( m \) is \( \tau_2 = m \times g \times d \). We need \( \tau_2 = 490 \text{ Nm} \). Substitute \( g = 9.8 \text{ m/s}^2 \) into the equation and solve for each option.

Evaluate each Option

1. **Option A:** \( m = 10 \text{ kg}, d = 2 \text{ m} \) \( \tau_2 = 10 \times 9.8 \times 2 = 196 \text{ Nm} \) (Not balanced)2. **Option B:** \( m = 50 \text{ kg}, d = 1 \text{ m} \) \( \tau_2 = 50 \times 9.8 \times 1 = 490 \text{ Nm} \) (Balanced)3. **Option C:** \( m = 40 \text{ kg}, d = 1.5 \text{ m} \) \( \tau_2 = 40 \times 9.8 \times 1.5 = 588 \text{ Nm} \) (Not balanced)4. **Option D:** \( m = 20 \text{ kg}, d = 2.5 \text{ m} \) \( \tau_2 = 20 \times 9.8 \times 2.5 = 490 \text{ Nm} \) (Balanced)

Determine the Correct Options

After evaluating the calculations, both options B and D provide a torque of 490 Nm, thereby balancing the teeter-totter. Thus, both options B and D balance the teeter-totter.

Key concepts

Seesaw balance

When we talk about balancing a seesaw, also known as a teeter-totter, we're diving into the intriguing world of physics, specifically the concept of torque. A seesaw is balanced when the torque on one side matches the torque on the other. Torque is a measure of how much a force acting on an object causes that object to rotate. In a seesaw, the pivot point (fulcrum) is the center, and our goal is to have equal torques on either side of this point.

• The seesaw will stay perfectly balanced when the counter-clockwise torque equals the clockwise torque.
• This occurs when the product of the force due to weight and the distance from the pivot is the same on both sides.
• In practical terms, this means if you have a heavier person on one side, they need to sit closer to the center, while a lighter person can sit further out.

To achieve balance on a seesaw, one can adjust either the mass sitting on it or the distance from the pivot, or both. Understanding this balance is crucial, especially in activities where stability is key.

Lever arm

A lever arm, in the context of a seesaw, is the distance from the pivot point to where the force is applied. This distance is crucial because the effect of a force depends not just on its magnitude but also on how far from the pivot point it acts.

• Think of the lever arm as the rod of the seesaw extending from the center pivot to either end.
• The longer the lever arm, the greater the torque, assuming the same amount of force is applied.
• This is why a child can lift an adult on a seesaw — if the child is far enough from the pivot.

Lever arms emphasize that not only the force is important but where you apply it matters too. For maximum effect, you want a long lever arm, especially when using a smaller force to balance out a larger force applied close to the pivot.

Force and distance relationship

The relationship between force and distance is foundational in understanding torque and balancing seesaws. In physics, this relationship is often controlled by the equation for torque: \[ \tau = F \times d \] Where \( \tau \) is the torque, \( F \) is the force, and \( d \) is the distance from the pivot point. This equation shows us:

• Increasing the force or the distance will increase the torque.
• Conversely, a smaller distance requires a larger force to produce the same torque, and vice versa.
• To balance a seesaw, the product of force and distance must be the same on both sides.

In exercises like the one provided, students learn to manipulate this relationship to find the right masses and distances to achieve a balanced seesaw. This understanding is not only theoretical but can be applied in everyday problem-solving scenarios where leverage is involved, such as using a crowbar or balancing scales.