Solve Modeling Higher-Order Diff Eqn: Free Undamped Motion w/ Spring

In summary, the mass is heading downward at a velocity of 3 ft/s after being released from the spring.
  • #1
Matthewmccoy6
2
0
A mass of 1 slug is suspended from a spring whose spring constant is 9lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of square root 3 ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s.

I found the differential equation to be x(t)=Acos(3t) + Bsin(3t).

What I did was I took downward as positive while upward as negative. I'm having trouble finding t.

My equation for the motion was x(t)= -cos 3t - (square root 3/ 3) sin 3t

Please help, thanks!
 
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  • #2


Matthewmccoy6 said:
A mass of 1 slug is suspended from a spring whose spring constant is 9lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of square root 3 ft/s. Find the times at which the mass is heading downward at a velocity of 3 ft/s.

I found the differential equation to be x(t)=Acos(3t) + Bsin(3t).
That is NOT a differential equation. I presume you mean you found that as the general solution to the differential equation.

What I did was I took downward as positive while upward as negative. I'm having trouble finding t.

My equation for the motion was x(t)= -cos 3t - (square root 3/ 3) sin 3t

Please help, thanks!
So you need to solve
[tex]-cos(3t)- (\sqrt{3}/3)sin(3t)= 3[/tex].
That is the same as
[tex]cos(3t)= 3- (\sqrt{3}/3)sin(3t)[/tex]

Squaring both sides of that gives
[tex]cos^2(3t)= 9- (2\sqrt{3}/3)sin(3t)+ sin^2(3t)[/tex]

Replace [itex]cos^2(3t)[/itex] with [itex]1- sin^2(3t)[/itex] and you have
[tex]1- sin^2(3t)= 9- (2\sqrt{3}/3)sin(3t)+ sin^2(3t)[/tex]
a quadratic equation you can solve for sin(3t) and then for t.

Be sure to check if an "extraneous" solutions were introduced by the squaring.
 
  • #3


I know, this is a differential equation problem though and using the equation of simple harmonic motion I found the above equation.
Also, I was hoping there was a simple identity that I could use to find t rather than something really unpleasant. Thanks though.
 

1. What is a higher-order differential equation?

A higher-order differential equation is a mathematical equation that involves derivatives of a function up to a certain order. In the context of solving motion equations, higher-order differential equations are used to describe the behavior of a system over time.

2. What is free undamped motion with a spring?

Free undamped motion with a spring is a type of motion where a mass is attached to a spring and allowed to oscillate without any external forces or damping. The motion is described by a second-order linear differential equation and can be solved using mathematical techniques such as the characteristic equation and initial conditions.

3. How do you solve modeling higher-order differential equations?

To solve modeling higher-order differential equations, you first need to identify the order of the equation and its type (linear or nonlinear). Then, you can use techniques such as substitution, separation of variables, or series solutions to solve the equation. You also need to consider any initial conditions to find the specific solution that describes the behavior of the system.

4. What is the significance of the spring constant in free undamped motion?

The spring constant, also known as the force constant, represents the stiffness of the spring and determines the strength of the force exerted on the mass. In free undamped motion, a higher spring constant will result in a higher frequency of oscillation, while a lower spring constant will result in a lower frequency of oscillation.

5. Can higher-order differential equations be applied to real-life situations?

Yes, higher-order differential equations have various applications in real-life situations, including physics, engineering, economics, and biology. They can be used to model the behavior of systems such as mechanical systems, electrical circuits, population growth, and chemical reactions.

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