What Is the Time Period of Oscillation for a Uniformly Stretched Spring?

[quawa99]

time period of a "SPRING"

Homework Statement

A spring has mass is m and natural length "L" and spring constant "k" has its one end fixed and the other end stretched by a length "a" and released.What is the time period of its oscillations?(there is no other mass attached to the spring,only the spring has mass m distributed uniformly across its length)

Homework Equations

Kx=ma
Energy of a stretched spring = 1/2kx^2+(kinetic energy)

The Attempt at a Solution

The net energy possessed by the system is constant(E).This energy exists in the form of kinetic and potential energy.
E=1/2(kx^2)+ kinetic energy

 

[adjacent]

Homework Statement

A spring has mass is m and natural length "L" and spring constant "k" has its one end fixed and the other end stretched by a length "a" and released.What is the time period of its oscillations?(there is no other mass attached to the spring,only the spring has mass m distributed uniformly across its length)

Homework Equations

Kx=ma
Energy of a stretched spring = 1/2kx^2+(kinetic energy)

The Attempt at a Solution

The net energy possessed by the system is constant(E).This energy exists in the form of kinetic and potential energy.
E=1/2(kx^2)+ kinetic energy

Where is your attempt?

 

[quawa99]

Where is your attempt?

I don't have any idea
I wanted to write the energy equation and defferentiate it with respect to time but I couldn't get the kinetic energy of the spring and relate it with x

 

[rude man]

A spring with mass is not an easy thing to analyze. Look for something on the Web.

 

[haruspex]

A spring with mass is not an easy thing to analyze. Look for something on the Web.

True, but that's because in general there can be wave motion within the spring. In this case, it starts uniformly stretched, and it's reasonably obvious (probably not hard to prove) that this will remain the case in subsequent motion.
quawa99, consider an element (relaxed) length ds of the spring at (relaxed) length s from the fixed end. Take the extension of the spring at some instant to be x. Assuming the spring is uniformly stretched at all times, what equations can you write for the forces on ds?