Dynamics of a Child Swinging: Applying Energy Conservation Principles

Understanding the dynamics of a child swinging on a swing is an excellent way to explore the principles of energy conservation in physics. This article will walk you through a step-by-step solution using the principle of conservation of energy to determine the speed of a 20 kg child at the bottom of a swing that hangs from 3 meters long chains. We will also explore how these principles can be applied in real-world scenarios.

Understanding the Scenario

Imagine a 20 kg child swinging on a swing that hangs from 3 meters long chains. The child swings out to a 45-degree angle before reversing direction. Our goal is to determine the speed of the child at the bottom of the arc, given the initial height of the angle.

Step-by-Step Solution

Step 1: Calculate the Height at the 45-Degree Angle

The height gained when swinging out to a 45-degree angle can be calculated using the length of the swing and the angle. The length of the swing, L 3.0 m.

[ h L - L cos(theta) ]

Where (theta 45^{circ}) and (cos(45^{circ}) frac{1}{sqrt{2}}).

Calculating ( h ):

[ h 3.0 text{ m} - 3.0 text{ m} cdot frac{1}{sqrt{2}} 3.0 text{ m} left(1 - frac{1}{sqrt{2}}right) ]

Calculating the numerical value:

[ h 3.0 text{ m} left(1 - 0.7071right) approx 3.0 text{ m} cdot 0.2929 approx 0.878 text{ m} ]

Step 2: Calculate the Potential Energy at the Top

The gravitational potential energy (PE) at the height ( h ) is given by:

[ PE mgh ]

Where ( m 20 text{ kg} ) (mass of the child), ( g 9.81 text{ m/s}^2 ) (acceleration due to gravity), and ( h approx 0.878 text{ m} ).

Calculating ( PE ):

[ PE 20 text{ kg} cdot 9.81 text{ m/s}^2 cdot 0.878 text{ m} approx 172.5 text{ J} ]

Step 3: Calculate the Kinetic Energy at the Bottom

At the lowest point, all potential energy is converted into kinetic energy (KE):

[ KE frac{1}{2} mv^2 ]

Setting ( KE PE ):

[ frac{1}{2} mv^2 172.5 text{ J} ]

Solving for ( v ):

[ frac{1}{2} cdot 20 text{ kg} cdot v^2 172.5 text{ J} ]

[ 10 v^2 172.5 ]

[ v^2 frac{172.5}{10} 17.25 ]

[ v sqrt{17.25} approx 4.15 text{ m/s} ]

Conclusion

The speed of the child at the bottom of the swing is approximately 4.15 m/s, demonstrating the effectiveness of energy conservation principles in real-world situations.

Real-World Applications

Understanding the dynamics of a child on a swing helps in various practical applications, such as designing playground equipment, improving safety measures, and promoting physical education. The principles of energy conservation are not only fundamental to physics but also have extensive applications in engineering, sports, and daily activities.

Key Concepts

Key concepts in this problem include:

Kinetic Energy (KE): Energy in motion Potential Energy (PE): Energy by virtue of an object's position Conservation of Energy: The total energy in an isolated system remains constant

Keywords

The keywords for this article are kinetic energy, potential energy, and conservation of energy.

References

For further reading and verification, refer to standard physics texts and online resources.