Simple Harmonic Motion: General Solution

47 Simple Harmonic Motion: General Solution

In the previous chapter we considered a mass on a spring that gets displaced an initial distance $d$ to the right. Using Newton’s laws, we found a very special differential equation: $\ddot x = -\omega^2 x$ , and found that the motion of the mass on the spring was described by the equation $x(t)=d\cos(\omega t)$ . That is, $x(t)=d\cos(\omega t)$ is a solution to the differential equation $\ddot x = -\omega^2 x$ . However, it is not the only solution.

Exercise 47.1: Another Solution

Newton’s laws for a mass on a spring result in the differential equation $\ddot x = -\omega^2 x$ where $\omega^2 = k/m$ . There, $k$ is the spring constant, and $m$ is the mass.

A. Verify that the equation $x(t)=A\sin(\omega t)$ is a solution to the differential equation.

B. In the previous chapter, we found a solution $x(t)=d\cos(\omega t)$ . This corresponded to an oscillating mass that starts its oscillation to the right of the equilibrium point, and at its maximum displacement. Here, we found that $x(t)=d\sin(\omega t)$ is also a solution. How do the oscillations start in this case?

C. Let’s drive home the point of this exercise with one last question.

The big take away here is this: the initial conditions of the problem tell you which solution you should use. This makes a lot of sense: if an object is oscillating back and forth, whether the oscillation is described by a sine or a cosine depends on when you “start” looking at the oscillation. Moreover, recall that

$\sin(\theta)=\cos(\theta+\pi/2)$
$-\cos(\theta)=\cos(\theta+\pi)$
$-\sin(\theta)=\cos(\theta+3\pi/2)$

In other words, we can think of all of our solutions as being a cosine solution, but with an angle $\phi$ added to it. That is,

The most general solution for simple harmonic motion is $x(t)=A\cos(\omega t+\phi)$ .

The angle $\phi$ is called the phase of the oscillation, and it is telling us where in the oscillation the problem starts (i.e. when $t=0$ ). For this book, we will only concern ourselves with oscillations that start either at maximum displacement (cosine solution) or with the object at the equilibrium point with some initial velocity (sine solutions), so our solution will always look like a pure sine or a pure cosine. Let’s try it.

Exercise 47.2: Spring Collision

A horizontal spring is attached to a wall. A mass $m$ slides on a frictionless table towards the spring with incoming speed $v_0$ . The mass collides with the spring and compresses it a distance $x_{\rm max}$ .

A. What is the velocity of the mass when the spring is fully compressed?

B. Find how much the spring is compressed.

I want to walk you through the solution. We start by describing what happens in English.

Now draw a picture. The story has a beginning and end: draw both, and label everything.

Now let’s use use our understanding of the story in English and our picture. We know the mass will move in simple harmonic motion. Determine the angular frequency of the oscillation, and whether the solution to this problem corresponds to $+\sin(\omega t)$ , $-\sin(\omega t)$ , $+\cos(\omega t)$ , or $-\cos(\omega t)$ . Use your answer to find the appropriate expression for $x(t)$ .

Our expression for $x(t)$ has an unknown amplitude. However, we know the initial velocity of the oscillations! Compute $v(t)$ , and set $v(0)=-v_0$ to find the amplitude of the oscillations, which is the answer to our original question.

Key Takeaways

47 Simple Harmonic Motion: General Solution

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