Solving Motion of a Particle: Types of Motion & Equation

[Gwilim]

Homework Statement

A particle of constant mass moves one dimensionally under the influence of a restoring force, proportional to the displacement from the origin, and a damping (resistive) force propportional to the velocity. Write down an appropriate equation of motion for the particle. escribe the three possible types of motion for this problem, describing the solution in each case.

2. The attempt at a solution

the equation of motion:

F = R - D

F is the resultant force, R is the restoring force, D is the damping force.

There are three constants in the problem: the mass of the particle and the constants of proportionality of the two forces. Calling them c1, c2 and c3:

c1x'' = c3x - c2x'

c1x'' + c2x' - c3x = 0

The three types of motion, at a guess, would be net acceleration, constant velcoity and net deceleration, but I have no good reason for suppoising this to be the case.

How do I complete the question?

 

[pgardn]

Do you have a problem in your textbook that deals with a mass attached to a spring on a horizontal table? hint restoring force...

Now do you have a problem that deals with friction as a function of velocity? hint maybe air resistance on the mass with the spring attached as stated above?

 

[gabbagabbahey]

[
There are three constants in the problem: the mass of the particle and the constants of proportionality of the two forces. Calling them c1, c2 and c3:

c1x'' = c3x - c2x'

c1x'' + c2x' - c3x = 0

First, the restoring force should be $-c_1 x$, so as to always accelerate the mass back towards the origin. Second, you really only need two constants, just divide your entire equation by the mass...$\frac{c_2}{c_1}$ and $\frac{c_3}{c_1}$ are just different constants, so you can write

$$x''+\alpha x' + \beta x=0$$

where $\alpha$ and $\beta$ are positive.

The three types of motion, at a guess, would be net acceleration, constant velcoity and net deceleration, but I have no good reason for suppoising this to be the case.

I would say that positive acceleration, zero acceleration and negative acceleration are 3 different phases of the mass' motion, not 3 different types of motion.

Instead, try solving the above ODE...you should find that there are 3 types of solutions depending on whether $\alpha^2-4\beta$ is equal to zero, less than zero or greater than zero. Physically these 3 solutions correspond to different types of damping...do you know what 3 types of damping these solutions correspond to?

 

[Gwilim]

First, the restoring force should be $-c_1 x$, so as to always accelerate the mass back towards the origin./

Thanks, I didn't knwo what a restoring force was, I assumed it meant that it restored the motion of the particle or some such nonsense.

Second, you really only need two constants, just divide your entire equation by the mass...$\frac{c_2}{c_1}$ and $\frac{c_3}{c_1}$ are just different constants, so you can write

$$x''+\alpha x' + \beta x=0$$

where $\alpha$ and $\beta$ are positive.

Good, thanks.

I would say that positive acceleration, zero acceleration and negative acceleration are 3 different phases of the mass' motion, not 3 different types of motion.

Yeah I thought my answer there was useless.

Instead, try solving the above ODE...you should find that there are 3 types of solutions depending on whether $\alpha^2-4\beta$ is equal to zero, less than zero or greater than zero. Physically these 3 solutions correspond to different types of damping

IIRC the three forms solutions to a second order ODE take are:

Ae^ax + Be^bx

Ae^ax + xBe^bx

Ke^kx(sin(ax)+cos(bx))

A B K a b k all being constants.

So I assume those are the three types of motion.

do you know what 3 types of damping these solutions correspond to?

I don't, no

 

[gabbagabbahey]

IIRC the three forms solutions to a second order ODE take are:

Ae^ax + Be^bx

Ae^ax + xBe^bx

Ke^kx(sin(ax)+cos(bx))

A B K a b k all being constants.

So I assume those are the three types of motion.

That's more or less correct, but you will have an easier time comparing these 3 types, by writing 'a' and 'b' in terms of $\alpha$ and $\beta$...pay special attention to the sign of the exponents in each case...which solutions decay the fastest?

Afterwards, look up the terms 'critical damping', 'underdamped' and 'overdamped'