Pure maths, physics, logic (braingames.ru): non-trade-related brain games - page 27

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MetaDriver:
Aha, your formula is about the same. Now think about it, from what term should the energy of vibration depend on stiffness and amplitude? I don't. Think again. It doesn't look like anything. It is known that the ball is perfectly elastic. That's enough. How exactly the waves walk in it, unlike the spring, is absolutely one-dimentional invariant - it does not influence on the amount of energy conserved in the vibrations.

There you go, written just above almost exactly the same:

Hence the total vibrational energy of the ball spring is:

E_vibr_ball = ( k*x^2 / 2 ) = M_brick * g*delta - m_ball*g*H / 4

 

So, I've got it like this:

m_ball = 2 * delta * M_brick / (1 - delta) ;

delta in metres

This is if we [correctly] assume that the energy of the ball after rebound is evenly distributed between vibrational and kinetic energy.

 

Yeah, it's a bit steep. But it has to be justified.

Here's the inequality from my last equation, which makes all this trouble possible in the first place:

M_brick / m_ball >= H / (4 *delta)

 
Mathemat:

Yeah, it's a bit steep. But it has to be justified.

Here's the inequality from my last equation, which makes all this trouble possible in the first place:

M_brick / m_ball >= H / (4 *delta)

I don't quite understand how it worked out, but not the point, I'll have another look.

I was thinking about stiffness. Only frequency and amplitude will depend on stiffness. But not energy of oscillations. It should be a constant.

// Well, that's my logic. Which, as we've found out, can be tricky.

 

The inequality comes out of the non-negativity of the vibrational energy:

Отсюда полная колебательная энергия пружины равна:

0 <= k*x^2 / 2 = M_brick * g*delta - m_ball*g*H / 4

I seem to have made a mess of the theorem: it's about the distribution of degrees of freedom between vibrational and rotational. It seems to have nothing to do with translational.

Something's missing.

 
Mathemat:

There's something a little missing.

I've been doing some mental experiments (I've been doing some experiments in my head).

For example. I imagined a spring in weightlessness freely released after compression against a wall. If you look at it slowly, it moves like a caterpillar. First it straightens out completely, then the rear starts to catch up, and the front almost(?) stops in the air until the spring is fully compressed again, then the cycle repeats. The centre of the spring then moves uniformly at V0/2

Which again leads me to the idea of a uniform distribution of energy between motion and oscillation...

 

There, I think I'm finally convinced. Stay tuned.

Let's return to the ball-spring idea. Now in the following form.

Cut an absolutely inelastic ball in half, insert (attention!) a weightless absolutely elastic spring inside.

Let's look at the moment of detachment: upper part of the ball (half of its mass) moves upwards with speed of brick, the other half stands motionless on the ground.

Then we get half the velocity of the motion. Obviously, the other half is eaten up by the oscillatory process.

Seems so convincing.

Any objections?

 

Unconvincing so far.

Далее получаем половинную скорость движения.  Очевидно что вторая половина съедена колебательным процессом.

Erm... you've made sure that half the speed is only a quarter of the energy. It's not halved.

I see the process this way: let the brick sink to its lowest point and compress the spring to its limit. Next, the spring starts to uncompress and accelerates the brick into space. When does the brick break away? At the point where the velocity of the spring is maximal, i.e. just at half of its distance to the maximum extension. This velocity is exactly equal to the initial velocity of the brick's flight into space.

On the other hand, you can try to estimate the total energy of a spring from that speed without touching its stiffness. Simply by the motion of its elementary masses. Anyway, it's something to think about. I myself have wondered how its energies are divided.

 
Mathemat:

Unconvincing so far.

Erm... you've made sure that half the velocity is only a quarter of the energy. It's not halved.

I see the process this way: let the brick sink to its lowest point and compress the spring to its limit. Next, the spring starts to uncompress and accelerates the brick into space. When does the brick break away? At the point where the velocity of the spring is maximal, i.e. just at half of its distance to the maximum extension. This velocity is exactly equal to the initial velocity of the brick's flight into space.

On the other hand, from that speed, you can try to estimate the total energy of the spring without touching its stiffness. Simply by the motion of its elementary masses. Anyway, it's something to think about. I myself have wondered how its energies are divided.

I did not find any contradiction. Not only that, it's finally cleared up, see:

E = (m/2)*Vbrick^2 + (m/2)*0^2 = m *(Vbrick/2)^2 + E

where E is the total energy of the ball-spring

(m/2)*Vbrick^2 is the energy of the upper half of the ball-spring at the moment of detachment of the brick

(m/2)*0^2 is the energy of the lower half of the ball-spring at the moment the brick comes off ( = 0 , of course)

m *(Vbrick/2)^2 is the kinetic energy of the rising spring ball

From which it follows that E-Vibrations = kinetic energy.

Ъ

Check.

// The easiest thing is to check exactly on my last "half-ball-half-spring" model. There is practically no chance of confusion, and no integrals.

// While the energy distribution is not affected by the device (construction) of the hopper.

 
Mathemat:

(5 points; who knows the answer - do not write!!!!)

Is it possible to arrange a regular tetrahedron in Cartesian coordinate system so that all its vertices lie at points with integer coordinates?

Somehow this stuff reminds me of Fermat's theorem in a poorly disguised form.
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