Understanding Longitudinal Wave Speed on a Spring: A Step-by-Step Approach

[rpcarroll]

This one has me pretty well stumped, I've tried several methods and I am basically getting tunnel vision now. Any help would be appreciated.

Homework Statement

(a) Show that the speed of a longitudinal wave along a spring of force constant 'k' is v=$$\sqrt{kL/\mu}$$, where L is the unstretched length of the spring and $$\mu$$ is the mass per unit length.

(b) A spring with mass .4kg has an unstretched length of 2m and a froce constant of 100 N/m. Use results of (a) to solve.

 

[jbsteele]

Homework Equations v=\sqrt{kL/\mu}The Attempt at a SolutionFor (a), we can use the equation v=\sqrt{kL/\mu} to prove this statement. This equation states that the speed of a longitudinal wave is equal to the square root of the force constant ‘k’ multiplied by the unstretched length of the spring, divided by the mass per unit length. This means that the speed of the wave is directly proportional to the force constant and the unstretched length, and inversely proportional to the mass per unit length.For (b), we can use the given equation to calculate the speed of the longitudinal wave. We are given the force constant (k=100 N/m), the unstretched length (L=2m), and the mass per unit length (\mu=0.4kg). Plugging these values into the equation, we get:v=\sqrt{100 \times 2 / 0.4} =\sqrt{500} =22.36 m/s.Therefore, the speed of the longitudinal wave along the spring is 22.36 m/s.

 

[thomate1]

As a scientist, it is important to approach problems with a clear and systematic method. In this case, we are dealing with the properties of longitudinal waves along a spring. Let us review the given information and break down the problem into smaller steps.

Firstly, we are given the equation v=\sqrt{kL/\mu} which relates the speed of the wave (v) to the force constant (k), unstretched length (L), and mass per unit length (\mu) of the spring. This equation is derived from the wave equation, which describes the relationship between the speed, wavelength, and frequency of a wave.

To solve part (a), we need to show how this equation is derived. This can be done by using the fundamental principles of physics, such as Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement from equilibrium. By using this law and Newton's Second Law of Motion, we can derive the wave equation and subsequently the speed equation.

For part (b), we are given specific values for the mass, unstretched length, and force constant of a spring. Using the equation from part (a), we can plug in these values to solve for the speed of the longitudinal wave along the spring.

In conclusion, it is important to approach problems in a step-by-step manner and utilize fundamental principles to understand and solve complex concepts. By breaking down the problem into smaller parts and using the given information, we can successfully solve for the speed of a longitudinal wave along a spring.