General Mechanics/Analysis Using Newton's Laws

Analysis Using Newton's Laws

The acceleration of the mass at any time is given by Newton's second law

$a={\frac {d^{2}x}{dt^{2}}}={\frac {F}{m}}=-{\frac {kx}{m}}$

An equation of this type is known as a differential equation since it involves a derivative of the dependent variable . Equations of this type are generally more difficult to solve than algebraic equations, as there are no universal techniques for solving all forms of such equations. In fact, it is fair to say that the solutions of most differential equations were originally obtained by guessing!

There are systematic ways of solving simple differential equations, such as this one, but for now we will use our knowledge of the physical problem to make an intelligent guess.

We know that the mass oscillates back and forth with a period that is independent of the amplitude of the oscillation. A function which might fill the bill is the sine function. Let us try substituting,

$x=A\sin(\omega t)$

where ω is a constant, into this equation.

We get

$-\omega ^{2}A\sin(\omega t)=-A{\frac {k}{m}}\sin(\omega t)$

Notice that the sine function cancels out, leaving us with $\omega ^{2}={\frac {k}{m}}$ . The guess thus works if we set

$\omega ={\sqrt {\frac {k}{m}}}$

This constant is the angular oscillation frequency for the oscillator, from which we infer the period of oscillation to be

$T=2\pi {\sqrt {\frac {m}{k}}}$

This agrees with the result of the dimensional analysis. Because this doesn't depend on $A$ , we can see that the period is independent of amplitude.

It is easy to show that the cosine function is equally valid as a solution,

$x=B\cos(\omega t)$

for the same $\omega$ .

In fact, the most general possible solution is just a combination of these two, i.e.

$x=A\sin(\omega t)+B\cos(\omega t)$

The values of $A,B$ depend on the position and velocity of the mass at time $t=0$ .