Springs
Springs are elastic devices that store and release energy through compression and elongation. They operate based on Hooke's Law, which states that the force exerted by a spring is proportional to its displacement from its equilibrium position. The spring constant (k) quantifies a spring's stiffness, with higher values indicating stiffer springs. When a spring is compressed or stretched, it accumulates potential energy, which can be converted into kinetic energy as the spring returns to its original shape.
Springs are fundamental components in various applications, from large industrial springs in railroad cars that enhance ride comfort to small, delicate springs in mechanical watches that ensure precision timekeeping. The behavior of springs also exemplifies concepts of harmonic oscillation, where a mass attached to a spring moves in a periodic manner, further illustrating the principles of energy conservation within mechanical systems. Understanding springs and their properties is essential in fields like physics, engineering, and mechanics, highlighting their versatility and importance in everyday technology.
Springs
FIELDS OF STUDY: Classical Mechanics
ABSTRACT: This article describes spring systems and their relation to Hooke’s law, the conservation of energy, and harmonic oscillators. Hooke’s law appears in all fields on physics and engineering because it can be applied to most solid objects in addition to springs. Harmonic oscillators are important systems in classical mechanics; they are found in nature and used for devices.
Principal Terms
- compression: the pushing forces applied to an object in order to diminish its size or volume.
- elongation: the lengthening of an object under stress.
- harmonic oscillations: the motion resulting from a system that when displaced from its equilibrium point exerts a restoring force that is proportional to the displacement.
- Hooke’s law: the principle that states that the force necessary to compress or stretch a spring by a certain displacement is proportional to that displacement.
- kinetic energy: the energy associated with motion.
- potential energy: stored energy due to position or configuration.
- spring constant: a mathematical value that defines the stiffness of a spring.
- tension: the pulling force exerted by the ends of a rope, wire, rod, or similar object.
Hooke’s Law
Elasticity is the ability of an object or material to return to its normal shape after experiencing elongation (lengthening) or compression (shortening) due to tension. An object is considered more elastic if it can be returned to its original shape more precisely. One example of an elastic object is the spring. When a spring is stretched or compressed, it exerts a restoring force that returns it to its beginning length. Hooke’s law states that this restoring force (F) is proportional to the amount (x) by which it is stretched or compressed:
F = kx
The spring constant (k) is a constant of proportionality that refers to the stiffness of the spring. The greater the value of k, the stiffer the spring. Force (F) has units of newtons (N), displacement (x) has units of meters (m), and therefore the spring constant (k) has units of newtons per meter (N/m). It is important to note that this equation gives magnitude only. To be more exact, one may establish the origin of the x-axis to be at the equilibrium length of a spring. If one stretches the spring in the positive x direction (x > 0), the spring exerts a force in the negative x direction with a magnitude of kx:
Fx = −kx
Likewise, if one compresses the spring in the negative x direction (x < 0), the spring exerts a force in the positive x direction with a magnitude of kx. The restoring force is opposite to the direction of the displacement. Hooke’s law only works for small stretches and compressions, however. If a spring is stretched excessively far, then it will reach a point where it will become permanently deformed and will not return to its original shape.
Hooke’s law can apply to forces other than those associated with springs. The force that holds atoms together can be modeled using Hooke’s law. These forces are responsible for vibrations and oscillations, normal force, and wave motion.
Energy of a Spring
Hooke’s law is also an example of the first law of thermodynamics. A spring conserves energy when it is compressed or stretched. The potential energy (U) of a spring when it has been displaced from its equilibrium by an amount x is:
U = ½kx2
The potential energy of a spring at its equilibrium point is zero and always positively increases as it is displaced from its equilibrium point. This displacement can be due to either stretching or compression. The potential energy of the spring can be converted to kinetic energy (K), both measured in joules (J). Kinetic energy depends on an object’s mass (m) and velocity (v):
K = ½mv2
Consider a spring with one end attached to a support and the other end attached to a mass. Now one compresses the spring by a certain distance, then releases it on a frictionless surface. The total energy (E) of this system is the sum of the potential and kinetic energy:
E = U + K
The mass oscillates back and forth, and the energy changes from potential to kinetic energy. The maximum potential energy occurs when the spring is at maximum compression and extension at the endpoints of the oscillation. At the endpoints, the kinetic energy is zero. The maximum kinetic energy happens when the mass travels through the equilibrium position. One can find the maximum velocity (vmax) of the mass by setting the maximum potential energy (Umax) equal to the maximum kinetic energy (Kmax) and solving for vmax:
Harmonic Oscillator
A spring with a mass attached to one end demonstrates simple harmonic motion and is a classic example of a harmonic oscillator. Consider a mass on a spring that has been stretched or compressed, then let go. The mass will then exhibit harmonic oscillations, moving back and forth about its equilibrium position. The period (T) of a mass on a spring is:
The unit for the period is the second (s). The period is independent of the amplitude and gravitational acceleration and is the same for horizontal and vertical spring systems.
Sample Problem
A spring is compressed 3 centimeters by an applied force of 75 newtons. Find the spring constant.
Answer:
Use Hooke’s equation to solve for the spring constant (k). Plug in the values for force (F) and displacement (x), remembering to convert displacement from centimeters (cm) to meters (m):
The spring constant is 2,500 N/m. By measuring the displacement of the spring from its original length by an applied force, one can determine the stiffness of the spring.
Springs in Everyday Use
There are many different types of springs that come in various sizes and shapes that provide different functions. Large springs used in railroad cars are heavy and stiff and used to smooth the ride of the car. Small delicate spiral springs are used in mechanical watches. The fact that these small springs obey Hooke’s law with a frequency determined by the mass and the spring stiffness allows for accurate mechanical watches and clocks to be manufactured.
Bibliography
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"Motion of a Mass on a Spring." Physics Classroom. Physics Classroom, n.d. Web. 4 July 2015.
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Williams, Matt, "What is Hooke’s Law?" Universe Today. Universe Today, 13 Feb. 2015. Web. 6 July 2015.