Question

At the local playground, a 16-kg child sits on the end of a horizontal teeter- totter, \(1.5 \mathrm{~m}\) from the pivot point. On the other side of the pivot an adult pushes straight down on the teeter-totter with a force of \(95 \mathrm{~N}\). In which direction does the teeter-totter rotate if the adult applies the force at a distance of (a) \(3.0 \mathrm{~m}\), (b) \(2.5 \mathrm{~m}\), or (c) \(2.0 \mathrm{~m}\) from the pivot? (Assume that the teeter-totter itself pivots at the center and produces zero torque.)

Short answer

(a) Towards child, (b) Towards child, (c) Towards adult.

Step-by-step solution

Determine Torque by the Child

Torque (\( \tau \) ) is calculated using the formula: \( \tau = r \cdot F \), where \( r \) is the distance from the pivot and \( F \) is the force applied. The gravitational force on the child is \( F = m \cdot g \), where \( m = 16 \mathrm{~kg} \) and \( g = 9.8 \mathrm{~m/s^2} \). So, \( F = 16 \times 9.8 = 156.8 \mathrm{~N} \). The distance is \( 1.5 \mathrm{~m} \). Thus, the torque by the child is \( \tau_{\text{child}} = 1.5 \times 156.8 = 235.2 \mathrm{~Nm} \).

Calculate Torque for Various Distances by the Adult

The torque (\( \tau \) ) applied by the adult is calculated with \( \tau_{\text{adult}} = r \cdot F \), and \( F = 95 \mathrm{~N} \).(a) With distance \( r = 3.0 \mathrm{~m} \), \( \tau_{\text{adult}} = 3.0 \times 95 = 285 \mathrm{~Nm} \).(b) With distance \( r = 2.5 \mathrm{~m} \), \( \tau_{\text{adult}} = 2.5 \times 95 = 237.5 \mathrm{~Nm} \).(c) With distance \( r = 2.0 \mathrm{~m} \), \( \tau_{\text{adult}} = 2.0 \times 95 = 190 \mathrm{~Nm} \).

Compare Torques and Determine Rotation Direction

Compare the torques when the adult applies force:(a) For \( 3.0 \mathrm{~m} \), \( 285 \mathrm{~Nm} > 235.2 \mathrm{~Nm} \); teeter-totter rotates towards child.(b) For \( 2.5 \mathrm{~m} \), \( 237.5 \mathrm{~Nm} > 235.2 \mathrm{~Nm} \); teeter-totter rotates towards child.(c) For \( 2.0 \mathrm{~m} \), \( 190 \mathrm{~Nm} < 235.2 \mathrm{~Nm} \); teeter-totter rotates towards adult.

Key concepts

Teeter-totter

The teeter-totter, also known as a seesaw, is a classic playground equipment that consists of a long board balanced on a central pivot point. This simple apparatus is a practical example of the principle of torque, or rotational force, in physics.
The design is straightforward but allows for fascinating interactions with physical forces based on the positioning and weight distribution of the users. On a teeter-totter, when one side goes up, the other automatically goes down.

This effect is largely due to how torque works around the pivot point. It is important to understand that the rotation of the teeter-totter is influenced by both the force applied and the distance from the pivot, which gives rise to the concept of moment or torque:

• The further a force is applied from the pivot, the greater the induced torque.
• Balance is achieved when the torques on both sides of the pivot are equal.

In any case where one of the users or objects on the teeter-totter has a greater torque, that side will move downward, causing the opposite side to rise.

Pivot point

The pivot point on a teeter-totter is the center spot or axis on which the board rotates. It's crucial in determining how the teeter-totter will balance. The pivot acts as the fulcrum in a lever system, which the teeter-totter essentially is.

To clarify, the pivot:

• Does not move; instead, it serves as the stationary point around which all rotational motion occurs.
• Devotes on the balance of torques, meaning adjustments in weight or force placement impact which end of the teeter-totter will lower or rise.

In problem-solving, calculating the effect of applied forces around the pivot requires understanding that the further the force is applied from the pivot, the bigger impact it has.
In practical terms, if you were to add weights or another child closer or farther from the pivot, it substantially changes how the teeter-totter behaves. Thus, strategic placement and calculation are fundamental in creating equilibrium.

Gravitational force

Gravitational force is a natural phenomenon by which all things with mass are brought towards one another. On Earth, it imparts all objects with a weight that can be calculated using the formula: \( F = m \cdot g \)
where \( F \) is the force in Newtons, \( m \) is the mass in kilograms, and \( g \) is the acceleration due to gravity, approximately \( 9.8 \, m/s^2 \).
This is the force that the child exerts on the teeter-totter, due to their mass.
When analyzing torque in problems like the teeter-totter, gravitational force is key. This force acts at the point where the child is sitting and contributes to the torque calculation by multiplying with the distance from the pivot. This multiplication results in the moment arm that can either stabilize the teeter-totter or cause it to rotate.
Essentially:

• Gravitational force's effect increases with the distance from the pivot.
• When this force causes torque greater than the opposing side, the teeter-totter will tip towards that side.

This interplay between weight, distance, and gravitational force exemplifies the basic principles of physics that govern balance and motion in playground equipment and beyond.