Solving Conceptual Problem: Mass Moving Downhill

[HWGXX7]

Hello,

I got into a very little and simple problem which I should understand. But I don't..

Look at this picture:

The mass naturally will move downwards from the hill. Oke, no problem. Analysing the external forces applied to this mass: gravity and friction.

Gravity, independent of time.
$$F_{g}=m.g$$

Friction force, depents on the speed the mass is moving down.
$$F_{w}=k.\frac{dx}{dt}$$

This I do understand.

Aply Newton second law: $$\sum F=m.a$$, this is a vectorial equation.
Means that the orientation of the forces is important, you also know this offcourse.

Now the tricky part I can't understand: I know de speed of the mass is getting smaller en smaller because de mass is slowing down because of friction.

Acceleration is defined as :$$a=\frac{dv}{dt}$$, in this case the derivative is negative.

So $$m.a$$ is also negative and must orient in the negative x direction...

I know I'am wrong but I can't find a decent reason in my logic for it..


gratefull for help

 

[huntoon]

sinθ

 

[HWGXX7]

@huntoon:

Sorry dude, but seems to me you don't understand what I ask and what the context is...
$$sin(\theta)$$ has nothing do to with this, even more I didn't specify any angle $$\theta$$

Somebody clearify my confussion?

thank you

 

[Rayquesto]

What's wrong with your thoughts? I don't see anything wrong. Perhaps, you are just wanting someone to elaborate on more dynamics of this situation. In that case, you would find the net force by knowing the angle and some trig. You know about it?

 

[HWGXX7]

Wow, I made u huge mistake: I confused the acceleration with the derivative of speed, instead of the second derivative of position x in function of time.The reference axis is position x and not speed! Because I don't know the equation of x(t) I cannot know the second derivative. But I do know that de mass is moving downward.



That why I couldn't get my logic clear..

In that case, you would find the net force by knowing the angle and some trig. You know about it?

Yes, I do know that the angle is needing for setting up de second order differetialequation, which wil solve the position vector x(t).

But if I wouls assume the direction of movement wrong, I cannot find the corretc equation for x(t), because the diffferentialequation would be wrong.

Correct?
So setting up differentiale equations demands:
- the forces with their correct orientation
- the resulting movement of the mass (or other object where Newton second law applies), suppose the mass is moving upward, thean I have include - sign to ma.

Is this a correct procedure to tackle a dynamic problem?
Thank for help.

 

[potatoecannon]

Hi HWGXX7,

Just to weigh in here on your conceptual problem; is this a special application you are trying to solve?

The usual formula for sliding (kinetic) friction is F_w=μ_sliding*N
where N is normal force, N = m * g * cos(angle).

This yields a constant friction force, which when coupled with the constant force of gravity, will result in a constant rate of acceleration (on a linear sloping surface).

 

[HWGXX7]

is this a special application you are trying to solve

No, as I stated in the title is was a conceptual problem.

Setting up differential equation demands accurate notition of sign convection, in this case: (using the reference axis as shown in the picture)

$$+m.\frac{d^{2}x}{dt^{2}}=-k.\frac{dx}{dt}+F_{g,x-component}$$
this gives me the correct solution for $$x(t)$$

But if I would write down:

$$-m.\frac{d^{2}x}{dt^{2}}=-k.\frac{dx}{dt}+F_{g,x-component}$$

gives me a wrong solution.

So it's important not only to pay attention to the direction of the know forces, but also sot the direction of the movement o f the particle itself.

grtz

 

[potatoecannon]

Alright sorry but I just wanted to check, since it's in the engineering section (instead of general physics).

There's nothing really to add; you seem to know the math pretty well & derived the correct equation.
It may have been slightly more confusing due to setting up a left-handed coordinate system?

But as for sign convention/directions, a good equation should be able to run backwards as well. (ie the slope changes from positive to negative, or the block is initially moving in a negative direction, etc).

The main thing is that the sign conventions match the relations you want to model; friction opposes motion (-), acceleration (+) is in the same direction as the gravity component (+).
A quick qualitative analysis is a good way to check for sign convention errors.

 

[HWGXX7]

The main thing is that the sign conventions match the relations you want to model; friction opposes motion (-), acceleration (+) is in the same direction as the gravity component (+).
A quick qualitative analysis is a good way to check for sign convention errors.

Yes indeed, one can choose a arbitrarily reference as long as the components in the whole equation are correct relative to the others.

The math I do know, took me while to get to this level and maintain it also. But I'am glad I do understand it, because once you get a taste of this, the road to understand the world of mechanical engineering is open.

Even the more that more than 99% of the people doesn't even have a clue what I talk about when I mention something trivial as Newton law...Which is in essence just the tip of the ice iceberg.

Thank anyway for responding ;)