How to Determine Equations of Motion for a String with a Mass at the Boundary?

In summary, the conversation discusses finding the equations of motion for a string with identical mass densities and tensions when an incident pulse travels from the left and is attached in the middle by a mass. The pulse is represented by y=Aexp[i(kx-wt)] and the mass acts as a moving boundary condition. The speaker is seeking help with determining the next steps, particularly in drawing a free body diagram to relate the slope of the string.
  • #1
wingnut290
1
0
Two strings with identical mass densities and tensions are attached in the middle by a mass m, an incident pulse is traveling from the left, I need to find the equations of motion for the string on the right.

The pulse is y=Aexp[i(kx-wt)]

work so far:
the mass will act as a moving boundry condition because of the verticle component of the string. So i have m(d^2y/dt^2)=-T dy/dt, i am having trouble in where to go next, can anyone help?
 
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  • #2
Draw an old fashioned FBD of the forces on the mass.
This will give you a BC relating dy/dx for each string.
 

FAQ: How to Determine Equations of Motion for a String with a Mass at the Boundary?

What are boundary conditions on a string?

Boundary conditions on a string refer to the specific constraints or limitations placed on the behavior of a string, typically in the context of physics or mathematics. These conditions dictate how the string can move or oscillate, and are crucial in determining the overall behavior of the system.

What is a fixed boundary condition on a string?

A fixed boundary condition on a string occurs when the ends of the string are fixed in place, meaning they cannot move or oscillate. This type of boundary condition is often used to simulate a string that is attached to two fixed points, such as a guitar string.

What is a free boundary condition on a string?

A free boundary condition on a string occurs when the ends of the string are free to move and oscillate. This is often used to simulate a string that is not attached to anything, such as a jump rope or a flag waving in the wind.

How do boundary conditions affect the behavior of a string?

Boundary conditions play a crucial role in determining the behavior of a string. They can affect the frequency, wavelength, and amplitude of the string's oscillations, as well as the overall shape of the string. Different boundary conditions can result in drastically different behaviors of the string.

What are some real-life applications of boundary conditions on strings?

Boundary conditions on strings have numerous real-life applications, particularly in the fields of acoustics and engineering. For example, understanding boundary conditions is crucial in designing musical instruments, such as stringed instruments like guitars and violins. They are also important in studying wave behavior in fluids and materials, and in analyzing the structural integrity of bridges and buildings.

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