Differential equation for motion in mass-spring system with impulse

In summary, the conversation discusses how an impulse affects the differential equation for motion in a mass-spring system. The equation governing the motion of the mass is non-homogeneous, with the impulse function being an additional function on the right side. The form of the homogenous differential equation is ay''+by'+cy=0, and the presence of the impulse changes it to ay''+by'+cy=g(x). The conversation also mentions the use of a hammer to provide an impulse and how it can be solved without this aid by using the homogenous equation.
  • #1
zpeters1
1
0
I am reviewing for a final and I don't know how an impulse affects the differential equation for motion in this mass-spring system. Can someone please help?

A mass m=1 is attached to a spring with constant k=2 and damping constant c=2. x(0)=0 & x'(0)=0. At the instant t=π, the mass is struck with a hammer, providing an impulse p=10.

Write the differential equation governing the motion of the mass.
 
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  • #2
Any external force applied to the object like that adds an additional function on the right side. You have a non-homogeneous equation with the impulse (delta) function being the additional function.
 
  • #3
Do you know the form of homogenous differential equation?

ay''+by'+cy=0

Now as impulse is also given in your question. It means it is an additional quantity. The question can be solve without this aid.
Then you use your homogenous equation of the form given above.
It implies that any additional quantity adds a function in the homogenous equation and make it a non homogenous equation as,

ay''+by'+cy=g(x)

First you try to write your homogenous equation according to your question.
 

What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives to represent the rate of change of a variable over time or space.

What is a mass-spring system?

A mass-spring system is a physical system that consists of a mass attached to a spring. When the mass is pulled or pushed, the spring is stretched or compressed, resulting in a force that causes the mass to oscillate around a fixed point.

What is an impulse in a mass-spring system?

In a mass-spring system, an impulse is a sudden change in the momentum of the mass. It can be caused by an external force or impact, and it results in a sudden displacement or change in velocity of the mass.

What is the differential equation for motion in a mass-spring system with impulse?

The differential equation for motion in a mass-spring system with impulse is given by m(d^2x/dt^2) + kx = F(t), where m is the mass, k is the spring constant, x is the displacement of the mass, t is time, and F(t) is the external force or impulse acting on the system.

How is the differential equation solved for motion in a mass-spring system with impulse?

The differential equation for motion in a mass-spring system with impulse can be solved using various methods, such as the Laplace transform, separation of variables, or numerical methods. The solution will depend on the initial conditions and the specific form of the external force or impulse acting on the system.

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