Spring Problem, Differential Equations

In summary, the conversation involves solving for the position of a mass attached to a spring and viscous damper, given initial conditions and the acceleration of gravity. The solution involves finding the damping constant and mass, setting up a differential equation, and solving for the constants c1 and c2 using the initial conditions. The final equation for the position of the mass is -1*e^(-2t)*sin(2t*sqrt(31))/sqrt(31), with careful attention to units.
  • #1
checkmatechamp
23
0

Homework Statement


A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper with a damping constant of 2 lb-s/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 2 in/s, find its position u at any time t. Assume the acceleration of gravity g = 32 ft/s2.

Homework Equations

The Attempt at a Solution



I solve for k and get 64, and solve for the mass and get 32/64, so my differential equation is 0.5y'' + 2y' + 64y = 0, I solve for r and get c1*e^(-2t)*cos(2t*sqrt(31)) + c2*e^(-2t)*sin(2t*sqrt(31))

My initial position is 0, so y(0) = 0, and my initial velocity is -2, so y'(0) = -2

So substituting, I get

0 = c1*e^0*cos(0) + c2*sin(0)

0 = c1

Now for y',

y' = -c1*e^(-2t)*sin(2t*sqrt(31))*2sqrt(31)) + -2c1*e^(-2t)*cos(2t*sqrt(31)) + c2*e^(-2t)*cos(2t*sqrt(31))*2sqrt(31) - 2*c2*e^(-2t)*sin(2t*sqrt(31))

-2 = -c1*e(0)*0 - 2c1e^(0)*cos(0) + c2*e^(0)*cos(0)*2sqrt(31) - 2*c2*e^(0)*sin(0))

-2 = -2c1 + c2*2sqrt(31)

But c1 is 0, so -2 = c2*2sqrt(31), and so c2 = -1/sqrt(31)

So my final equation is -1*e^(-2t)*sin(2t*sqrt(31))/sqrt(31)

But when I pick that as an option, the computer marks it wrong. I see some options with a 12sqrt(31) on the bottom, but I don't think that's it.
 
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  • #2
checkmatechamp said:

Homework Statement


A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper with a damping constant of 2 lb-s/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 2 in/s, find its position u at any time t. Assume the acceleration of gravity g = 32 ft/s2.

Homework Equations

The Attempt at a Solution



I solve for k and get 64, and solve for the mass and get 32/64, so my differential equation is 0.5y'' + 2y' + 64y = 0, I solve for r and get c1*e^(-2t)*cos(2t*sqrt(31)) + c2*e^(-2t)*sin(2t*sqrt(31))

My initial position is 0, so y(0) = 0, and my initial velocity is -2, so y'(0) = -2

So substituting, I get

0 = c1*e^0*cos(0) + c2*sin(0)

0 = c1

Now for y',

y' = -c1*e^(-2t)*sin(2t*sqrt(31))*2sqrt(31)) + -2c1*e^(-2t)*cos(2t*sqrt(31)) + c2*e^(-2t)*cos(2t*sqrt(31))*2sqrt(31) - 2*c2*e^(-2t)*sin(2t*sqrt(31))

-2 = -c1*e(0)*0 - 2c1e^(0)*cos(0) + c2*e^(0)*cos(0)*2sqrt(31) - 2*c2*e^(0)*sin(0))

-2 = -2c1 + c2*2sqrt(31)

But c1 is 0, so -2 = c2*2sqrt(31), and so c2 = -1/sqrt(31)

So my final equation is -1*e^(-2t)*sin(2t*sqrt(31))/sqrt(31)

But when I pick that as an option, the computer marks it wrong. I see some options with a 12sqrt(31) on the bottom, but I don't think that's it.

You have to be careful with units here. If the initial velocity is -2 in/s, then you can't just blindly plug y'(0) = 2 into your equation, because the damping constant was given as 2 lb-s/ft. The initial velocity should be -2/12 ft/s, to convert in/sec to ft/sec.
 

1. What is a "Spring Problem" in the context of Differential Equations?

The "Spring Problem" is a classic mathematical problem that involves finding the motion of a spring system that is subject to external forces and a damping force. It is often used to model real-world phenomena such as oscillating motion in mechanical systems.

2. What are Differential Equations and how are they used in the Spring Problem?

Differential Equations are mathematical equations that involve the derivatives of an unknown function. In the context of the Spring Problem, they are used to model the behavior of the spring system by relating the displacement and velocity of the object to the external forces and damping force acting on it.

3. What is the general form of the Differential Equation used to solve the Spring Problem?

The general form of the Differential Equation used to solve the Spring Problem is known as the "Harmonic Oscillator Equation." It is a second-order linear differential equation that can be written as mx'' + bx' + kx = 0, where m is the mass of the object, b is the damping coefficient, and k is the spring constant.

4. How do you solve the Spring Problem using Differential Equations?

To solve the Spring Problem using Differential Equations, we first set up the Harmonic Oscillator Equation and then use techniques such as the method of undetermined coefficients or the Laplace transform to solve for the displacement function. We can then use this function to find the velocity and acceleration of the object at any given time.

5. What are some real-world applications of the Spring Problem and Differential Equations?

The Spring Problem and Differential Equations have many real-world applications, such as modeling the vibrations of musical instruments, analyzing the motion of a pendulum, and studying the behavior of electrical circuits. They are also used in fields such as engineering, physics, and economics to understand and predict the behavior of complex systems.

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