Question

Two children are on adjacent playground swings of the same height. They are given pushes by an adult and then left to swing. Assuming that each child on a swing can be treated as a simple pendulum and that friction is negligible, which child takes the longer time for one complete swing (has a longer period)? a) the bigger child d) the child given the b) the lighter child biggest push c) neither child

Short answer

Answer: (c) Neither child has a longer period.

Step-by-step solution

Understand what a simple pendulum is

A simple pendulum consists of a small, heavy object (the “pendulum bob”) suspended by a light, inextensible string. The motion of the pendulum is determined by the length of the string and the angle it makes with the vertical, called the displacement angle.

Derive the equation for the period of a simple pendulum

The equation for the period T (the time it takes for one full oscillation) of a simple pendulum can be derived from the equation of motion using the small angle approximation. The equation for the period of a simple pendulum is given by: T = 2\pi\sqrt{\frac{L}{g}} where L is the length of the pendulum, and g is the acceleration due to gravity.

Identify the relevant parameters from the exercise

The exercise asks about the period of a pendulum that is affected by two factors: the mass of the child and the initial push given. According to the equation in step 2, the period T of a simple pendulum depends only on the length L and acceleration due to gravity g. The mass of the child and the initial push are not included in the equation.

Determine which child has a longer period

Since the period of a simple pendulum does not depend on its mass or the initial push given, both children will have the same period, assuming they are on swings of the same height (length). Thus, neither child has a longer period. #Conclusion# The correct answer is (c): neither child has a longer period, as the period of a simple pendulum depends only on its length and the acceleration due to gravity, and not on the mass of the child or the initial push given.

Key concepts

Pendulum Period

The period of a simple pendulum refers to the time it takes for the pendulum to complete one full swing, going from one side to the other and back. It's an essential concept in understanding pendulum motion and is determined by the physical properties of the pendulum itself.
In the context of a simple pendulum, the period \( T \) is given by the formula:\[ T = 2\pi\sqrt{\frac{L}{g}} \]where:

• \( L \) is the length of the pendulum.
• \( g \) is the acceleration due to gravity, typically about \( 9.81 \text{ m/s}^2 \) on Earth.

Interestingly, this formula shows that the period of a simple pendulum is independent of the mass of the pendulum bob or the initial push given. Therefore, two pendulums of equal length will have the same period, regardless of their masses or how hard they are swung. This principle simplifies the understanding of pendulum motion significantly, as you only need to consider the length of the pendulum to predict its period.

Small Angle Approximation

In the study of pendulums, especially simple pendulums, the small angle approximation is a useful concept. It states that for small angles (usually less than 15 degrees), the sine of the angle is approximately equal to the angle itself when measured in radians. This makes the mathematics involved much simpler.
For a pendulum, the angle refers to the initial angular displacement from the vertical position. If this angle is small, we can use the small angle approximation to simplify the equations of motion. Specifically, we get:
\[ \sin(\theta) \approx \theta \] for small \( \theta \) (in radians).
This approximation allows us to derive the expression for the period of a pendulum without needing complex trigonometric functions. It's worth noting that as the angle of displacement increases, this approximation becomes less accurate, potentially affecting the precision of the period calculation slightly.

Equation of Motion

The equation of motion for a simple pendulum describes how it moves over time. It is derived from Newton's laws of motion and involves variables like the acceleration due to gravity, the length of the pendulum, and the angle of displacement.
The fundamental differential equation for a simple pendulum is:\[ \frac{d^2\theta}{dt^2} + \frac{g}{L} \sin(\theta) = 0 \]where:

• \( \theta \) is the angular displacement.
• \( t \) represents time.
• \( g \) is the acceleration due to gravity.
• \( L \) is the length of the pendulum.

Using the small angle approximation, this equation simplifies to the simple harmonic motion form:
\[ \frac{d^2\theta}{dt^2} + \frac{g}{L} \theta = 0 \]
This equation resembles that of other harmonic oscillators like springs, making it easier to study pendulums using established methods. The solution to this equation helps determine how a pendulum's angle changes with time, ultimately leading to understanding the pendulum's behavior and predicting its period accurately.