Diff eq, Spring/mass damped/driven-Am I doing this right?

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In summary, the conversation discusses solving equations of motion for a damped and driven spring/mass system. The first problem involves finding the equation of motion, amplitude, period, and times at which the weight passes through equilibrium position for a 10lb weight attached to a 5ft spring with resistence and downward velocity. The second problem involves finding the equation of motion for a 1 slug mass attached to a spring with a damping force and an external force of 8sin4t applied, as well as determining the amplitude, period, and times at which the weight passes through equilibrium position.
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Diff eq, Spring/mass damped/driven--Am I doing this right?!

Homework Statement


1.) After a 10lb weight is attached to a 5ft spring, the spring measures 7ft. The 10lb is replaced with 8lb, and placed in a medium offering resistence equal to the instantaneous velocity.
A) find the equation of motion if the weight is released 0.5ft below equilibrium with downward vel of 1ft/s.

B) What are the amplitude and period of the motion? How many complete vibrations does the weight complete in [tex]2\pi[/tex] seconds?

C) Find the times at which the weight passes through the equilibrium position heading downward.


2.) A mass of 1 slug, when attached to a spring, stretches it 2ft and then comes to rest in the equilibrium position. Starting at t=0, an external force equal to f(t)=8sin4t is applied to the system. Find the equation of motion if the surrounding medium offers a damping force numerically equal to 8 times the instantaneous velocity.

Thanks..

The Attempt at a Solution



1.)
e3p1.jpg


2.) I left it with C1 and C2 b/c no initial cond were given.
e3p2.jpg
 
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  • #2
A) mx" + kx + cv = F(t) mx" + 8v + 8x = 8sin4t B) Amplitude: ? Period: 2\pi/4 Complete vibrations: \pi/2 C) t1=0, t2=2\pi/4
 

FAQ: Diff eq, Spring/mass damped/driven-Am I doing this right?

1. What is a differential equation?

A differential equation is an equation that relates a function to its derivatives. It is commonly used to model continuous change and is widely used in various fields of science such as physics, engineering, and economics.

2. What is a spring/mass system?

A spring/mass system is a physical system that consists of a mass attached to a spring. The mass is able to move freely along a horizontal axis, while the spring exerts a restoring force on the mass. This system is commonly used to model various real-life phenomena such as oscillations and vibrations.

3. What does it mean for a spring/mass system to be damped?

A damped spring/mass system is one in which the amplitude of oscillations decreases over time due to the dissipative effects of friction or resistance. This can be represented mathematically by adding a damping term to the differential equation that governs the system.

4. How do you solve a spring/mass system with a driving force?

To solve a spring/mass system with a driving force, you need to first set up the differential equation that describes the system. Then, you can use various techniques such as separation of variables, variation of parameters, or Laplace transforms to solve the equation and find the solution for the position of the mass as a function of time.

5. How can I check if I am solving a spring/mass system correctly?

To check if you are solving a spring/mass system correctly, you can compare your solution to known solutions or use physical intuition to verify if your solution makes sense. Additionally, you can check your calculations and ensure that all the steps are correct and consistent.

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