Harmonic Motion: How Does a Particle Behave Under a Sinusoidal Force?

[superdave]

Homework Statement

1. A particle of mass m is constrained to move along a straight line. In a certain
region of motion near x = 0 , the force acting on the particle is F = -F_0 sin(bx) , where F_0 and b are positive constants.

(a) Find the potential energy of the particle in this region. Sketch this potential; label the axes.

(b) If at time t = 0 the particle was at x = 0 and had velocity v_0 , find the turning points of theparticle’s motion.

(c) Find the period of the particle’s small harmonic oscillations about the equilibrium point.
What condition must be satisfied for the oscillations to be “small”?

Homework Equations

F = - dV/dx

The Attempt at a Solution

Part a)
So I get V = -int(F) = - F_0/b cos (bx). But I'm confused about the usual + C you get when doing an integral. I guess that would be V_0 which would be - F_0/b cos (b * x_0)? So does V = -F_0/b cos (bx) - F_0/b cos (bx_0)? And in that case, wouldn't it be pretty impossible to sketch without knowing x_0 which isn't defined in this part of the problem?

part b)

Umm, okay. so I get a=-a_0 sin(bx) = dv/dt. v = integral (a dt) = -a_0 * t * sin(bx) + v_0
So this confuses me, so I guess turning points are when t*sin(bx) = v_0/a_0? That's kind of a random answer and doesn't sit well.

part c)

I have no idea

 

[turin]

The reference point of potential energy is arbitrary; you can choose it to be whatever you want. I strongly suggest that you review the derivation of gravitational potential energy and/or electrostatic potential energy (both linear approximation and also inverse square law), and review the derivation of the potential energy of a simple harmonic oscillator (e.g. a mass on a spring). There are "standard" reference points, and these examples should give you an idea (especally the SHO, ;) ).

For part (b), think about what happens to the different energies (at the turning points).

BTW, if I'm not mistaken, your problem actually describes a pendulum in disguise.

 

[superdave]

Oh, I guess I was going about it wrong. I was looking at when v=0 for turning points, but I should be looking at when U is maximum.

So when x = integer multiples of pi/b. But then, why was that other stuff put in the problem about v_0? And that doesn't help me with part c.

 

[turin]

Maybe you're confusing vee's. There are two of them: capital V means potential energy; lower-case v means velocity. These are completely different. Maybe use capital U for potential energy to save confusion.

For part (c), give me something more to work with.

 

[superdave]

No, I wasn't confusing them. part b says

(b) If at time t = 0 the particle was at x = 0 and had velocity v_0 , find the turning points of the particle’s motion.

But I can find the turning points using U when it is maximized, as you indicated. When x=pi/b * integer. So the problem has a bunch of extraneous information?

c) Find the period of the particle’s small harmonic oscillations about the equilibrium point.
What condition must be satisfied for the oscillations to be “small”?

I know it's small when bx can be substituted for sin(bx).

T = (2*pi)/sqrt(k/m) is all I can think of. but then how to get k?

 

[superdave]

thanks for helping, btw. I'm just stressed because my midterm is tomorrow and I can't even do these problems from the practice test and I can't afford to fail this test or this this class. But I just can't seem to get it.

 

[superdave]

OOOH, can it be that A = pi/b (the turning point)! And A=v_0*T/2pi so T=2pi^2/(b*v_0)

Can that be right?

What kind of professor doesn't post the solution to the practice exam? sigh.

 

[turin]

I know it's small when bx can be substituted for sin(bx).

Good. But, if this is on your test, then your prof is probably also wanting you to justify this approximation. In other words, say something about sin(bx)-bx, or something about the next term in the Taylor series.

T = (2*pi)/sqrt(k/m) is all I can think of. but then how to get k?

That's not how I would do it, but that will work. Make that small-angle substitution in the force equation, and then compare to Hooke's law.