Introduction:

In this article, we will explore a world dealing with vibrations and waves, unveiling the secrets of how waves propagate and move. We will start by understanding the fundamental properties of vibrational motion and connect them to the concept of waves, allowing us to comprehend how energy is transmitted through these fascinating phenomena. You will learn how waves spread and interact with the surrounding environment, describing the behavior of waves in multiple ways. Your learning won't be limited to theories only; you will gain a practical understanding of the significance of these phenomena in daily life. Through our study of these phenomena and concepts, we will grasp the importance of understanding the behavior of waves and vibrations in various fields. We will delve into understanding resonance phenomena and the potential harmful effects if not well understood. Additionally, we will take a look at how bridges and structures are built safely using the principles of vibrations and waves. To understand how science can open countless doors to comprehend and explain the world around us, we embark on this exciting journey that will leave you amazed and enlightened by the end!

Article Elements:

1.       How do waves propagate in a spring?

2.       Periodic Motion:

3.       The Mass on a Spring:

4.       Simple Pendulums:

5.       Resonance:

6.       Conclusion

How do waves propagate in a spring?

How do pulses sent through a spring move when its other end is fixed? Waves in a spring propagate through the sequential transfer of particles within the material that forms the spring. When the spring is compressed or stretched at a specific point, it transfers this change in pressure or expansion to the neighboring particles. This change causes a wave of pressure or expansion to travel through the spring. If the other end of the spring is fixed, the wave will similarly pass through the spring. In this case, the spring will not move as a whole, but the wave will travel through it without the spring itself moving. This is similar to how waves travel on the surface of water when you touch it with your finger. The particles in the water move periodically, and this is related to the transfer of energy through the material without the entire material moving.

Periodic Motion:

Perhaps you've seen a pendulum clock swinging back and forth, noticing that each swing follows the same path, and each round trip takes the same amount of time. This motion is an example of periodic motion. Other examples include the oscillation of a metal body attached to a spring upward and downward. These motions that repeat in a regular cycle are examples of periodic motion.

In all these examples, the body is at a single position where the net force acting on it is zero, and the body is in equilibrium. When the body is pulled away from its equilibrium position, the net force becomes non-zero, and this force works to bring the body back towards the equilibrium position. If the force that brings the body back to its equilibrium position is directly proportional to the displacement of the body, the resulting motion is called simple harmonic motion. Two quantities describe simple harmonic motion: the period T, which is the time the body takes to complete one full cycle of back-and-forth motion, and the amplitude of oscillation, which is the maximum distance the body moves away from the equilibrium position.

The Mass on a Spring:

How does the spring interact with the force acting on it? Consider a spring with a mass hanging from its end without any external force acting on it. In this position, the spring does not stretch because there is no external force affecting it. The figure illustrates the spring hanging with a body of weight mg, and the spring has elongated by a displacement, balancing the upward force of the spring with the downward force of gravity. This elongation or compression of the spring corresponds to the displacement of 20, with the weight at its end being twice the weight in the previous equilibrium position (2mg). This aligns with Hooke's Law, stating that the force exerted by a spring is proportional to its elongation, and springs that follow this condition are called elastic springs and satisfy Hooke's Law, expressed as follows:

Hooke's LawF=-KX

The force exerted by a spring equals the product of the spring constant and the displacement it elongates or compresses from its equilibrium position.

In this equation, the spring constant depends on the stiffness of the spring and other properties, and x represents the displacement elongated or compressed by the spring from its equilibrium position.

Potential Energy:

When a force acts to elongate a spring, such as hanging a body at its end, there is a linear relationship between the applied force and the elongation of the spring. The slope of the graph represents the spring constant, measured in N/m. The area under the curve represents the work done to elongate the spring, which is equal to the elastic potential energy stored in the spring due to this work. The base of the triangle represents the displacement, and the height of the triangle represents the force equal to x according to Hooke's Law. Thus, the elastic potential energy stored in the spring is expressed by the following equation:

Elastic Potential Energy in a Spring   

The elastic potential energy in a spring equals half the product of the spring constant and the square of its displacement.

The unit of elastic potential energy is "N.m" or Joules (J).

How does the net force depend on the position? When a body is suspended at the end of a spring, the spring elongates until the upward force of the spring balances the weight of the body. At this point, the body is in its equilibrium position. If the hanging body is pulled downward, the force of the spring increases, producing a net force upward equal to the force of your pull plus the weight of the body. When the hanging body is released, it accelerates upward, and as the body moves upward, the elongation of the spring decreases, resulting in a decrease in the force directed upward.

The force exerted by the spring upward becomes equal to the weight of the body, resulting in a net force of zero. As a result, the system does not accelerate, and the body continues its upward motion above the equilibrium position. The net force is opposite to the direction of the body's displacement and is directly proportional to it. Therefore, the body moves in simple harmonic motion and returns to its equilibrium position, as shown in the figure.

Simple Pendulums:

Simple harmonic motion can also be illustrated through the swinging motion of a pendulum. A simple pendulum consists of a dense solid body called the pendulum bob suspended from a string of length 1. When the pendulum bob is pulled to one side and released, it oscillates back and forth, as shown in the figure. The string exerts tension force (F) on the pendulum bob, and gravity also affects the bob with a force (F). The vector sum of these two forces represents the net force. In the right and left positions in the figure, the net force on the pendulum bob and its acceleration are maximized, while the velocity is zero. In the middle position (equilibrium) in the same figure, the net force and acceleration are zero, while the velocity is maximized.

You can observe that the net force is a restoring force, always opposite to the direction of the pendulum bob's displacement, working to bring the bob back to its equilibrium position. When the angle of deviation of the string is small (approximately less than 15%), the restoring force is directly proportional to the displacement. This motion is then called simple harmonic motion. The period of the pendulum is calculated using the following equation:

Pendulum Period     

The period of the pendulum equals 2π multiplied by the square root of the quotient of the pendulum string length and the gravitational acceleration.

Note that the period of a simple pendulum depends only on the length of the pendulum string and the gravitational acceleration, not on the mass of the pendulum bob or the amplitude of the oscillation. In practical applications of the pendulum, it is used to calculate g (acceleration due to gravity), which varies slightly from location to location on the Earth's surface.

Resonance:

To set a swing in motion while sitting on it, push it by bending backward and pulling the rope (or chain) from the same point in each swing, or have a friend push you at regular intervals. Resonance occurs when small forces act on an oscillating or vibrating body at regular time intervals, leading to an increase in the amplitude of the vibration or oscillation. The time interval between the application of force on the vibrating body is equal to the period of the vibration. Common examples of resonance include rocking a car back and forth to free its wheels from sand when stuck, repetitive jumping on a diving board, or scuba diving. The large amplitude resulting from resonance may cause a feeling of stress.

Resonance is a distinctive form of simple harmonic motion, where small increases in the force magnitude at specific times during the body's motion lead to larger increases in displacement. Resonance resulting from the wind's motion, for example, aligning with the design of bridge supports, may be the cause of the collapse of the Tacoma Narrows Bridge.

Conclusion:

1.       We have learned about how waves propagate in springs and how springs interact with the forces acting on them.

2.       We explored periodic motion and simple harmonic motion and how springs can be used to explain these motions.

3.       We introduced the concept of elastic potential energy and how it relates to the level of forces and elongation in the spring.

4.       Given the importance of springs in many applications and mechanical systems, it is crucial to study and understand these phenomena in depth.

5.       Understanding the principles of periodic motion and springs is key to understanding many natural and technological systems.

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