- #1

Kernul

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## Homework Statement

A mass ##m## on a frictionless table is connected to a spring with spring constant ##k## so that the force on it is ##F_x = -kx## where ##x## is the distance of the mass from its equilibrium position. It is then pulled so that the spring is stretched by a distance ##x## from its equilibrium position and at ##t = 0## is released.

Write Newton’s Second Law and solve for the acceleration. Solve for the acceleration and write the result as a second order, homogeneous differential equation of motion for this system.

## Homework Equations

Newtons's Second Law.

Hooke's Law

Differential Equations

## The Attempt at a Solution

I write the the Newton's Second Law and solve for the acceleration:

$$F = m a_x = - k x$$

$$a_x = -\frac{k}{m} x$$

Now it tells me to write the result as a second order, homogeneous differential equation of motion. I don't quite get how I should do this but I think this way:

I write ##a_x = -\frac{k}{m} x## as ##\frac{d v_x}{d t} = -\frac{k}{m} x## and multiplying both sides for ##d t## and integrating I have:

$$v_x(t) = -\frac{k}{m} x t + C$$

where ##C## is a constant and would actually be ##v_{0x} = 0##

Same thing again with ##\frac{d x}{d t} = -\frac{k}{m} x t## and having:

$$x(t) = -\frac{k}{2m} x t^2 + C$$

Now, should I put all this like

$$x''(t) + x'(t) + x(t) = 0$$

and so

$$(-\frac{k}{m} x) + (-\frac{k}{m} x t) + (-\frac{k}{2m} x t^2) =0$$

Is this the correct way?