Question

A schoolyard teeter-totter with a total length of \(5.2 \mathrm{~m}\) and a mass of \(38 \mathrm{~kg}\) is pivoted at its center. A 19-kg child sits on one end of the teeter-totter. Where should a parent push vertically downward with a force of \(210 \mathrm{~N}\) in order to hold the teeter-totter level?

Short answer

The parent should push at approximately 2.38 m from the pivot.

Step-by-step solution

Identify the Pivot Point

The teeter-totter is pivoted at its center, meaning the pivot point is at the midpoint, which is at a distance of \( \frac{5.2}{2} = 2.6 \mathrm{~m} \) from either end.

Analyze Forces Acting on the Teeter-totter

The forces acting on the teeter-totter include the gravitational force due to the child, the weight of the teeter-totter, and the downward force by the parent. The child's force will act at one end, and the parent's force will act at a position \( x \) m from the center.

Calculate the Torque Due to the Child

Torque due to the child is calculated using the formula \( \text{torque} = ext{force} \times ext{distance} \). Here, \( ext{force} = 19 \text{ kg} \times 9.8 \text{ m/s}^2 \) and \( ext{distance} = 2.6 \text{ m} \). Hence, the torque is \( 19 \times 9.8 \times 2.6 \text{ Nm} \).

Calculate the Torque Exerted by the Parent

The parent applies a force of \( 210 \text{ N} \) at a distance \( x \) from the pivot. The torque due to this force is \( 210 \times x \text{ Nm} \).

Ensure Teeter-totter is in Equilibrium

For the teeter-totter to be level, the clockwise and counter-clockwise torques must be equal. Thus, the equation \( 19 \times 9.8 \times 2.6 = 210 \times x \) must be satisfied.

Solve for x

Rearranging the equation, we find \( x = \frac{19 \times 9.8 \times 2.6}{210} \text{ m} \). Calculating this gives \( x \approx 2.38 \text{ m} \).

Key concepts

Understanding the Pivot Point

The pivot point is a crucial aspect of balance in systems like teeter-totters. Think of it as the fulcrum or the central point about which the structure rotates or balances. In our example, the teeter-totter is pivoted exactly in the middle. This midpoint divides the entire length into two equal halves, each measuring 2.6 meters. This positioning ensures that any forces exerted on the teeter-totter will cause it to rotate around this pivot point. Understanding where this point is helps in calculating how different forces influence the balance.

Gravitational Force and Weight on the Teeter-totter

Gravitational force is the force with which the earth attracts a body towards itself. On the teeter-totter, this force factors into the weight of the child and the structure itself. To calculate the gravitational force acting on the child sitting at one end, we use the formula: \( F = m imes g \), where \( m \) is the mass of the child and \( g \) is the acceleration due to gravity, approximately \( 9.8 \text{ m/s}^2 \). For our 19-kg child, the force becomes \( 19 \times 9.8 = 186.2 \text{ N} \). This force, acting downward, creates a torque that attempts to tip the teeter-totter over the pivot point.

Force Calculation and Torque

Forces and the resulting torques they create are pivotal to understanding balance. Torque is the rotational equivalent of linear force and determines how much a force can cause an object to rotate about the pivot point. The formula for torque is: \( \text{Torque} = \text{Force} \times \text{Distance to pivot} \). The child's force exerts a torque calculated as \( 186.2 \text{ N} \times 2.6 \text{ m} = 484.12 \text{ Nm} \). For equilibrium, the parent's force should balance this torque. The force applied by the parent (210 N) will be adjusted using the same torque formula to find the position \( x \) from the pivot where it must be applied.

Achieving Equilibrium Condition

Equilibrium in this context means the teeter-totter is perfectly balanced. For equilibrium to occur, the sum of the torques around the pivot point must be zero. This means that all clockwise torques (like that from the child's weight) must be canceled by the counterclockwise torques (like that from the parent pushing downwards).The equation for equilibrium is: \( 19 \times 9.8 \times 2.6 = 210 \times x \). Solving for \( x \), we rearrange and calculate \( x = \frac{19 \times 9.8 \times 2.6}{210} \), which results in approximately 2.38 meters. This result indicates the exact position where the parent's force should be applied to maintain balance, ensuring that the teeter-totter remains level.