A spring of mass \(m\) and spring constant \(k\) has an unstretched length \(L_{0} .\) Find an expression for the speed of transverse waves on this spring when it's been stretched to a length \(L\)

At the heart of understanding how springs behave under force is Hooke's Law. Simply stated, Hooke's Law tells us that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. In the language of physics, it is expressed as
\[ F = -kx \].
In this formula,
\( F \) represents the force in newtons (N),
\( k \) represents the spring constant, measured in newtons per meter (N/m), and
\( x \) is the displacement in meters (m). The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement.
When a spring is stretched, the restoring force that tries to bring it back to its original length is what causes tension. This tension can be used in turn to create motion, such as causing a mass on the spring to oscillate, or in our case, to propagate a wave along the length of the spring. The relationship described by Hooke's Law is fundamental not only for understanding springs but also for analyzing any system where restoring forces play a role, such as in simple harmonic motion and waves.

The spring constant, denoted as \( k \), is key in determining how a spring will respond to applied forces. It's a measure of the spring's stiffness: the higher the spring constant, the stiffer the spring, and hence the greater the force required to cause a particular displacement. Mathematically, we can say:
\( k = \frac{F}{x} \),
where
\( F \) is the force applied and
\( x \) is the displacement from the spring's original length.
In the context of our exercise, the spring constant tells us how much tension the spring generates when stretched. This tension, which is the force along the length of the spring, is what supports the propagation of transverse waves. Thus, the spring constant is a crucial parameter in calculating the wave speed on the spring.

Linear mass density, represented by the Greek letter \( \mu \), is a property of any object that can be thought of as a line, such as a string, a rod, or a spring. It describes how much mass the object has per unit length. To calculate it, you simply take the total mass of the object and divide it by its length:
\[ \mu = \frac{m}{L} \].
Where
\( m \) is the mass, and
\( L \) is the length over which the mass is distributed. In our exercise, the spring's linear mass density plays a critical role because it affects how quickly waves can travel along the spring. The lighter the spring per unit length (the lower the \( \mu \)), the faster a wave can move. Understanding linear mass density is not only important in the context of waves on a spring but also for waves on strings, such as in musical instruments, and is an essential concept in the study of wave mechanics.