Periodic Force Action: Can It Displace a Particle Without Integration?

[Bob_for_short]

It is not a home work.

Let us suppose that at t=0 a particle is at rest. At t=0 we switch on a periodic force F(t) = F₀sin(ωt). Without integrating the Newton equation, do you think such a force is capable of displacing the particle to very far places as the time goes on?

 

[Dale]

Yes. I think a cosine would not.

 

[Bob_for_short]

Yes. I think a cosine would not.

Right answer, congratulations!
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[DocZaius]

What is the thinking behind this solution? Was it reached mathematically (but without integrating the Newton equation) or intuitively?

 

[Dale]

Kind of a little of both. During the positive lobe of the sine wave the object gains momentum. Then during the negative lobe it loses momentum. The amount of momentum gained is equal to the amount lost (effectively an integration in my head) so the velocity is always positive or zero, never negative.

 

[DocZaius]

It is not a home work.

Let us suppose that at t=0 a particle is at rest. At t=0 we switch on a periodic force F(t) = F₀sin(ωt). Without integrating the Newton equation, do you think such a force is capable of displacing the particle to very far places as the time goes on?

Although I infer (perhaps wrongly) that you mean to say the force would be rotating and thus changing direction periodically, it seems from looking at your equation that the force will merely be scaled with time, and not redirected. Even though there is an angular velocity in the sin function, the sin function still results in a scalar. Could you specify which assumptions I wrongly made or more likely, should have made?

edit: Upon thinking it over, I think you do mean it as a scalar and a one dimensional problem. In that case I suppose it does matter if the problem starts with maximum force (cosine) or with minimum force (sine)...

 

[Bob_for_short]

Exactly! It is a 1D problem. It happens that the initial phase of the force is important. Only in the particular case of cos(ωt) the particle oscillates around the initial position. In any other case the particle "drifts" away.