Physics problem

[Andrez]

I'm stuck on this, can anybody help me out? It's like I'm missing a link...

A mass m = 0.02 kg slips down a sloped plane (smooth, no friction). At the end of the slope it slips across a path with friction (ud = 0.1) and finally it hurts a spring fixed to a vertical wall, which at rest it's L0 = 10 cm and a constant of k = 2 N / m. The distance between the end of the slope and the wall is d = 40 cm. Considering that at t = 0 the mass v = 0, we need to know:

a - the height needed to make the mass hit the spring with a force that allows the mass itself to touch the wall

b - the height the mass reaches when it bounces back from the wall

Here's a simple draw of the problem:

[DeafieGamer]

Add elephant in the way.

[Orynge]

I forgot the formulae. I'm revising physics today and I'll post here if I find the right ones.

[Luc@s]

Its all about energy.
While the mass slips down its gravitational energy turns into kinetic energy, till it reaches the end of the slope. This part of the motion is conservative (no friction) which means the total energy is conserved.
Then the mass travels along a path where friction is not neglected anymore which means it loses energy. The energy lost during its process is only the work of the friction force (which is easy to calculate considering the assumptions made).

Before hitting the wall, the mass has to compress the spring entirely, which means it has to transform some of its kinetic energy into elastic energy. So the energy initially put in the system (due to the gravitational potential) minus the energy lost by friction has to be greater than the energy needed to press the spring, in order for the mass to hit the wall.

After that, the mass will bounce and a similar reasoning will give you the answer for b. But be careful, as it is not obvious that the mass will have enough energy left to reach the slope.

do you need help to write the expressions of these different energies ?

[Mousikos]

I'm guessing Ud=Coefficient of Kinetic friction?

[Andrez]

Music: we use to call it 'dynamic friction', as it's different from 'static friction' (us), but I guess you mean that.

Luc@s, thanks, that's just a nice start, but yes, I'd need some formulas too. The main thing that has been bugging me up is the fact I don't have the angle of the slope, so I figured out that I must do everything in 'reverse'.

So, the energy produced by the spring should be
Espring = 1/2*k*(L-L0)^2

assuming the block must touch the wall, the final length (L) of the spring must be 0, so Espring would be:
Espring = 1/2*2*(-0.01)^2 = 10^-4

Now, the kinetic energy of the block:
Ek = 1/2*m*v^2

And? How do I proceed here? v would be known if I knew the slope part, but I'm apparently stuck in a thing that should be easy as hell.

EDIT: perhaps a nice approach would be thinking of the of the block Ek = -Espring, Ek = 10^4 J, that is the energy needed to compress the spring at the limit. So:
10^4 = 1/2*m*v^2 ---> v^2 = 10^4/2*m = 0.0025
v = sqrt(v^2) = 0.05 m/s = 5 cm/s

According to this, the initial speed after the slope would be 5 cm/s, or 0,05 m/s, which would have sense, but I feel something is still wrong

[Luc@s]

The energy needed to compress the spring is indeed
but 10 cm = 0.1 m ;) (

You should proceed step by step. First, what is the total energy E when the mass is dropped ? Its not moving at this very moment so K = 0. But it has some gravitational energy. If we choose the reference so that the gravitational energy down the slope is 0, then .

When the mass gets down the slope, the total energy is still the same as before () because there was supposedly no friction. The only difference is that the energy was entirely transformed under its kinetic form (E = K), because the mass reached a position where its potential energy is 0.

The energy required to move along the path is obtained from the work of the friction force which is now taken into account. This force is constant during the motion. Thus, so that at the end of the path

(F can be calculated from the weight of the object and the friction coefficient)

Now, the kinetic energy has to be greater than the elastic energy required (which you calculated already).

[Andrez]

Arf, sorry for that mistake, sometimes I'm on blackout!

Anyway, I thought of using mgh, but stupid of me, I didn't think the slope was not to be taken into account since there is no friction at all!

Thanks Lucas, now it's clear! ;)

[Luc@s]

The slope is taken into account as it is the source of energy ;) (mgh, later partially transformed in elastic energy)
Note that the reason why you can avoid to solve the equations of motions (working only with energies) is because all the forces involved are "conservative" in this example. The solid friction is of course not conservative per se but it behaves mathematically as if it was derived from a potential which does not depend explicitly on time. So at the end some quantities are conserved and it makes the problem relatively easy to solve.
It would be more complicated if air friction was taken into account as well.

[Marti]

I shouldve paid more attention in physics class :/

[Andrez]

Indeed the slope is taken into account, but I meant that I thought mgh was only to be used in case of a freefalling object. I guess a freefall and sliding down through a slope must be the same thing in a mathematical context (considering both with no friction), and slope just changes the fact that instead of a block "sunk" into terrain (along y-axis), we'll have a mass with kinetic energy that goes on along the x-axis.

[Mousikos]

Andrez, draw a force diagram....

[Luc@s]

reasoning with forces is counter productive here