Question

At the local playground, a \(16-\mathrm{kg}\) 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) Clockwise, (b) Clockwise, (c) Counterclockwise.

Step-by-step solution

Understanding the Problem

To solve this problem, we need to consider the torque effect on both sides of the teeter-totter. Torque is the rotational equivalent of force and is calculated by multiplying the force exerted with the distance from the pivot point: \( \tau = F \times d \). The child contributes to the counterclockwise torque, while the adult contributes to the clockwise torque. We need to compare these torques to determine the direction of rotation.

Calculate Child's Torque

For the child, the force is due to gravity, \( F = m \cdot g = 16 \cdot 9.8 = 156.8 \text{ N} \), and the distance is \( 1.5 \text{ m} \). Thus, the child's torque is \( \tau_{ ext{child}} = 156.8 \times 1.5 = 235.2 \text{ Nm} \) counterclockwise.

Situation (a): Calculate Adult's Torque at 3.0 m

The adult applies a force of \( 95 \text{ N} \) at \( 3.0 \text{ m} \) from the pivot. Calculate the torque: \( \tau_{ ext{adult}} = 95 \times 3.0 = 285 \text{ Nm} \) clockwise. Compare with the child's torque (235.2 Nm). The adult's torque is greater, so the teeter-totter rotates clockwise.

Situation (b): Calculate Adult's Torque at 2.5 m

The adult's force at \( 2.5 \text{ m} \): \( \tau_{ ext{adult}} = 95 \times 2.5 = 237.5 \text{ Nm} \) clockwise. Comparing with 235.2 Nm from the child, the adult's torque is slightly greater, so the teeter-totter rotates clockwise.

Situation (c): Calculate Adult's Torque at 2.0 m

The adult's force at \( 2.0 \text{ m} \): \( \tau_{ ext{adult}} = 95 \times 2.0 = 190 \text{ Nm} \) clockwise. Comparing with 235.2 Nm from the child, the child's torque is greater, so the teeter-totter rotates counterclockwise.

Key concepts

Teeter-totter

A teeter-totter, also known as a seesaw, is a simple and classic playground apparatus that consists of a board balanced on a central pivot point. It allows for children to experience fun and play while also subtly introducing them to the concepts of balance, weight distribution, and basic physics principles. The teeter-totter works by leveraging the concept of rotational motion about a fixed point.
In a perfect world, a teeter-totter is balanced when the torques (the rotational forces) on each side are equal. This device is crucial for understanding how various forces and distances can influence motion and balance.

• If a child sits at one end, the weight and the distance from the pivot contribute to the torque.
• Another force, such as an adult pushing the other side, impacts balance and rotation direction.

The teeter-totter is often the first physics lesson for many children, showcasing the interaction of forces in a simple, yet effective way.

Torque calculation

Torque is a measure of how much a force acting on an object causes it to rotate. In simple terms, torque is the rotational equivalent of linear force. The formula used to calculate torque is: \[ \tau = F \times d \]where \(\tau\) represents torque, \(F\) is the force applied, and \(d\) is the perpendicular distance from the pivot point.
The magnitude of torque tells us how strong the rotational force is, while the distance from the pivot point determines the lever arm, which can significantly affect the motion.

• Large forces closer to the pivot may have the same torque as smaller forces further away.
• Rotational direction can be clockwise or counterclockwise, depending on the direction of the force applied.

Understanding how to calculate torque is essential for predicting how objects will spin and balance on devices like teeter-totters.

Pivot point

The pivot point is the central balancing position on a teeter-totter. It is the critical spot where the rotation occurs. Imagine it as the fulcrum in physical terms, providing the foundation upon which the plank rotates. Its placement determines how easily the board can tip either way.
The pivot point holds the teeter-totter in place and allows for precise motion. Properly analyzing the role of the pivot point is essential for understanding balance. It separates the forces exerted on either side and aids in calculating torques:

• Determining whether forces cause a clockwise or counterclockwise rotation depends heavily on their distance from the pivot.
• The closer a force is applied to the pivot, the less torque it generates, since the lever arm is shorter.

Recognizing and working with the pivot point helps us understand why certain forces are more effective than others in creating rotational motion.

Counterclockwise and clockwise rotation

Rotational direction plays a crucial role in analyzing the dynamics of a teeter-totter. The direction in which an object rotates is described as either clockwise or counterclockwise.
In our everyday experience:

• Clockwise rotation is when a force turns objects to the right, similar to the clock's hands.
• Counterclockwise rotation turns objects to the left, opposite of the clock's hands motion.

In scenarios involving torques, like our teeter-totter exercise, identifying which way the teeter-totter turns based on the applied forces is vital.
When an adult applies force:

• If the applied torque on one side exceeds the opposing side's torque, the board will rotate in the direction of the larger torque.
• Determining the rotation direction helps us predict which end will rise or fall, influencing the teeter-totter's balance.

Understanding the differences between these two rotational motions is key to figuring out movement patterns in rotational systems.