Equation of Motion for a Particle Under a Linear Restoring Force

[davesface]

Homework Statement

Find the equation of motion for a particle of mass m subject to a force F(x)=-kx where k is a positive constant. Write down the equation of motion as x''(t)=F/m. Then show that x(t)=Ce^iwt is a solution to the equation of motion for any C as long as w has one of 2 possible values (i is the imaginary unit, w is omega, t is time). What are those values?

There's more to it, but I am totally lost as to how I can at least start from this information.

Homework Equations

x''(t)=F/m
F(x)=-kx, where k is a positive constant
x(t)=Ce^iwt

The Attempt at a Solution

I took the derivative of the last equation listed in b twice to get x'(t)=iwCe^iwt and then x''(t)=i²w²Ce^iwt, which simplifies to x''(t)=-w²Ce^iwt.

I guess that I really would just like to know if I'm anywhere in the ballpark for how the problem should begin. It's not a graded problem, but I hate leaving it unsolved.

 

[HallsofIvy]

Homework Statement

Find the equation of motion for a particle of mass m subject to a force F(x)=-kx where k is a positive constant. Write down the equation of motion as x''(t)=F/m. Then show that x(t)=Ce^iwt is a solution to the equation of motion for any C as long as w has one of 2 possible values (i is the imaginary unit, w is omega, t is time). What are those values?

There's more to it, but I am totally lost as to how I can at least start from this information.

Homework Equations

x''(t)=F/m
F(x)=-kx, where k is a positive constant
x(t)=Ce^iwt

The Attempt at a Solution

I took the derivative of the last equation listed in b twice to get x'(t)=iwCe^iwt and then x''(t)=i²w²Ce^iwt, which simplifies to x''(t)=-w²Ce^iwt.

I guess that I really would just like to know if I'm anywhere in the ballpark for how the problem should begin. It's not a graded problem, but I hate leaving it unsolved.

Looks to me like you are doing the problem backwards! You are first asked to write down the equation of motion. You give as "relevant equations" x"= F/m and F= -kx. Okay, looks to me like the equation of motion is x"= -kx/m.

NOW you can argue that if x= Ce^iwt, then x'= Ciwe^iw and x"= -Cw²e^iwt= -w²(Ce^iwt which is the same as -kx/m as long as w²= -k/m. That last equation should tell you what values w can have.

 

[davesface]

x"= -Cw²e^iwt= -w²(Ce^iwt which is the same as -kx/m as long as w²= -k/m. That last equation should tell you what values w can have.

2 questions there:
1. Why is Ce^iwt which is the same as -kx/m as long as w²= -k/m?
2. How does w²= -k/m lead me to the values of w?