Sum of torque

neufneufneuf

Not sure what you mean.

Without the black arm: Disks are increasing their kinetic energy because w1 > w2, w2(t)=w2+Frt/I2. If w1=0, disks reduce their kinetic energy, true ?

With the black arm: If I set w1=0 and w0=+10 (for example), disks are increasing their energy but like w1=0, the term -Frtw1 is very low, no ? Maybe w1 must be not exactly at 0 but lower than before, just for reduce the term -Frtw1. I'm not sure about the rotationnal velocity, maybe it's w1' I need to decrease. If w1'=0 => w1=w0. The torque from disks to cruz is apply to w1' or w1 ?

NB the exact expression with 4 disks is:

[latex]8(-Frt\omega_1[/latex]+[latex]4\frac{F^2r^2t^2}{I_1}[/latex]+[latex]Frt\omega_2[/latex]+[latex]\frac{F^2r^2t^2}{2I_2})[/latex]

Edit:

Why not just get rid of it?

I would like to reduce w1 (the rotationnal velocity of the cruz) for reduce -Frtw1 and keep a rotation w0 (the arm) for increase rotationnal velocity of disks when there is friction between disks.

dlouth15

neufneufneuf wrote: »

Edit: I would like to reduce w1 (the rotationnal velocity of the cruz) for reduce -Frtw1 and keep a rotation w0 (the arm) for increase rotationnal velocity of disks when there is friction between disks.

But having the black arm whatever its angular velocity has no effect on w1, the angular velocity of the cross.

neufneufneuf

The torque from disks to the cruz is apply on w1 or w1' ?

dlouth15

neufneufneuf wrote: »

The torque from disks to the cruz is apply on w1 or w1' ?

In English what is w1'? What does it represent?

neufneufneuf

The torque from disks to the cruz works with the rotationnal velocity of w1 (rotationnal velocity of the cruz in the labo frame reference) or in w1' (rotationnal velocity of the cruz in the arm frame reference) ?

dlouth15

neufneufneuf wrote: »

The torque from disks to the cruz works with the rotationnal velocity of w1 (rotationnal velocity of the cruz in the labo frame reference) or in w1' (rotationnal velocity of the cruz in the arm frame reference) ?

Both. Because if you vary one you vary the other. But I think we are only interested in absolute rotations because rotational kinetic energy depends on absolute rotational velocities. I don't see a need for relative rotational velocities in this problem. If the sum of energy is zero with the cross and the disks on their own then the problem is solved.

If you add in the rotating black arm, then since there is no torque between the black arm and the cross, the velocity of the black arm remains constant and therefore its contribution to kinetic energy is constant. Therefore it doesn't play a part in the problem. And nor does a frame of reference that rotates with the black arm.

neufneufneuf

Case A
1/ turn counterclockwise the ring at w2, need the energy [latex]\frac{1}{2}mr^2w_2^2[/latex], with m the mass of the ring
2/ turn the ring with arm clockwise at w1, need the energy[latex]\frac{1}{2}md^2w_1^2[/latex], with d the length of the arm
3/ eject small part of ring when point are like green points, eject all part of the ring, recover kinetic energy [latex]\frac{1}{4}m((d[/latex]+[latex]r)w_1[/latex]-[latex]rw_2 ) ^2[/latex]+[latex]\frac{1}{4}m((d-r)w_1[/latex]+[latex]rw_2)^2[/latex]



Case B
1/ turn clockwise the ring at w2, need the energy[latex]\frac{1}{2}mr^2w_2^2[/latex]
2/ turn the ring with arm clockwise at w1, need the energy [latex]\frac{1}{2}md^2w_1^2[/latex]
3/ eject small part of ring when point are like green points, eject all part of the ring, recover kinetic energy [latex]\frac{1}{4}m((d[/latex]+[latex]r)w_1[/latex]+[latex]rw_2)^2[/latex]+[latex]\frac{1}{4}m((d-r)w_1[/latex]-[latex]rw_2)^2[/latex]



The energy needed for turn the ring around itself and with the arm is the same in case A and case B. In the contrary, the energy recover in case A is not the same than in case B.

neufneufneuf

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Eject 2 masses

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In this case, I don't break the ring. I add 2 objects for have a clockwise torque on the ring.

The ring is turning at w2 and the arm is turning at w1 with w2 < w1, I take 2 external blue objects (free to move in space, with mass m_o) that are turning at w1, like the ring turns counterclockwise in the arm reference, it's possible to eject these 2 objects and apply a clockwise torque to the ring in the same time. The two external objects don't lost the energy from the rotation w1, in the contrary the velocity after shock increase the velocity of free objects and the device win energy from the ring. Like this:



Blue objects keep Vw1 the velocity from w1 and have additionnal rotationnal velocity Vw2:



The energy needed at start is :

[latex]\frac{1}{2}mr^2w_2^2[/latex]+[latex]\frac{1}{2}md^2w_1^2[/latex]+[latex]2\frac{1}{2}m_od^2w_1^2[/latex]

with:
m: mass of the ring
[latex]m_0[/latex]: mass of free object
d: length of arm
r: radius of ring

The energy after "shock":

[latex]\frac{1}{2}mr^2w_3^2[/latex]+[latex]\frac{1}{2}md^2w_1^2[/latex]+[latex]2\frac{1}{2}m_od^2w_4^2[/latex]

with :

w3 > w2
w4 > w1

neufneufneuf

The arm is turning clockwise at w1 and the ring is turning at w2, with w2 < w1. With one external object, free to move in space, that move in a straight line at the velocity V:



There is a shock at time = 1, IN HORIZONTAL AXIS (the shock is not in vertical axis), between the blue object and the ring.

The trajectory of the ring is more at left than at right:

[latex]rsin((w2-w1)t)[/latex]

compare at:

[latex]d-\sqrt(r^2+d^2)cos(artg(\frac{r}{d})-w_1t)[/latex]

Forces are like that:



The horizontal shock gives F1 to the free object, and gives F2 to the ring. F3 is apply because there is F2, F3 don't give a torque to the arm. F4 is the force on the ring, F2/F4 gives a clockwie torque to the ring.

For example, with:

r=1
d=3
w2=0
w1=10
t=1e-5

the result is :

9.9999e-5 at left
9.9985e-5 at right

So the shock move at left the free object and increases the kinetic energy of the ring. The arm don't receive a torque because the force is in the axis.






Algodoo shows the trace of the point, it is moving at left:



Maybe a better position for the shock is:




I did a video:

https://www.youtube.com/watch?v=HWZBj_WiPHA

neufneufneuf

The black arm is turning clockwise at w1. A motor accelerates more and more a slave disk (inertial disk). I guess w1 constant because I recover the energy from the black arm for keep constant w1. The belt don't receive a torque, all forces Fa, Fb, Fc, and Fd cancel themselves. The motor receives -F (Fm1+Fm2 in vector) but this force is in the axis, so no torque from -F. The slave receives forces F1 and F2, the sum of F1 and F2 is F, so the black arm receives a clockwise torque +FR . The energy won is +FRw1t. The energy gave from the motor is in the kinetic energy of the slave, so this energy is not lost, it can be recover later.



The belt give the force only when I drawn them (need a special belt):

neufneufneuf

The last message with the belt seems correct. But with a belt it's difficult to have forces like I drawn, it's possible to use an electromagnet disk over the motor for replace the belt (electromagnets are not a motor, it is only something for replace the belt). The motor and the slave disk are turning clockwise like w1. The motor need the energy for accelerate the disk, but FRtw1 is not from the motor it come from the geometry. The motor is turning clockwise at w1 and accelerates after. Electromagnets give a counterclockwise to the motor but the motor take energy and accelerate. An external system not drawn recover energy from the black arm like that w1 is constant.

neufneufneuf

I would like study this basic device:

No gravity. No friction. Only one motor. An external device not drawn recover energy from the black arm for have w1 constant. At start, all objects are turning at w1. The goal is to recover the torque NFR when I use N basic systems. For that, the motor need to accelerate more and more, if the rotationnal velocity of the motor is constant the force F doesn't exist. But the energy is not lost it is in the inertial disk.



Ok, I guess the energy is conserved in this case. But I can place 2 basic devices:



In this case the torque on the black arm is 2FR, the disk increases its kinetic energy but in the same value than with one device. The energy from the motor is the same. There is only one motor, and one inertial disk. At the middle, it is 2 rings, with a mass as lower as possible.

I can use N basic devices:



The torque is NFR. The inertial disk increases its kinetic energy with the same value than in the case 1/ .

The belt must PUSH one side and PULL other side, this need a special belt.


The bigger ring must turn at w. The smaller turns at R1/R2w, with R1 the radius of the bigger ring and R2 the radius of the smaller ring.

Like an induction motor:



Maybe I can use an epicyclidal train :https://www.youtube.com/watch?v=2lk9yyki9RI , look at time=42 s and after

The belt can be replace by a hydraulic pipe and use turbines at the left part.

neufneufneuf

I understood why a basic device don't work, the motor gives a counterclockwise torque on the support. If I compare the energy with w1=0 and the energy with w1=100 it is the same. I simplified the basic device:



Remember, the motor accelerates more and more, never stop. With all forces:



An example:

With R1=3, R2=1, I=1, F=1

w1=0

The inertial disk is increasing its rotationnal velocity from 0 to 1. The motor needs to give te energy [latex]\frac{1}{2}I[/latex]. Note the time for goes from w=0 to w=1 is FR2/I=1.

w1=100

The velocity of the disk goes from 100 to 101 so the energy of the disk is increasing of [latex]\frac{1}{2}I(100[/latex]+[latex]1)^2[/latex]=[latex]100[/latex]+[latex]\frac{1}{2}[/latex]. But the motor gives a counterclockwise torque FR1, this decreases the energy of the device of 3*1*100*1=300. The torque from {F5,F6} gives the energy [latex](R1-R2)Fw_1t=2*1*100*1=200[/latex]. At final the motor gives 1/2, but the support lost 100 and the disk won 100, the sum is well at 0.

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Like I know where come from the energy lost by the torque F5,F6. I can test my device with a lot of basic devices in serial:



The disk will increase its velocity like before : +1 rd/s. The disk win the same energy +100 J. The motor gives the same torque on the support: -300 J. But the torque from F5,F6 is multiplied by N, with N the number of basic systems.

I don't need a special belt for push and pull. Just a tooth belt. I need 2 specials surfaces for assure that force from the belt are applied only when I drawn them.


The only thing I need to be sure it's the system for reduce the rotationnal velocity of the bigger ring. It musn't apply a torque on the support.

I can use 3 gears for each intermediate rings:



The device becomes:



Gears must have a mass as lower as possible. Like that the motor give a torque only to the terminal inertial disk. So if the disk is the same in my last example, with R1=3, R2=1, I=1, F=1, the disk win 100 the support lost 300 from the torque from the motor. Each intermediate device win 200 to the support, the sum is +100-300+3*200=400 J. Remember an external device recover energy from the support like that w1 is constant.

For a continous mouvement, it's possible replace the inertial disk by a brake. Like that the disk don't accelerate more and more, the support receive an additionnal clockwise torque from the brake that can be recover by an external device on the main axis (red).

A basic devices with gears reduce the velocity with a rapport of 3.33.

If I want keep w I can do that:



Instead to accelerate more and more, I take a brake, like that all rotationnal velocities are constant, w and w1 :



w1 is constant, an external device recover energy from the red axis. The brake can be an electromagnetic device like that the energy is recover, don't waste is in heating.

neufneufneuf

Another idea for the reductor:

neufneufneuf

A simple device like this:



On a support that is turning clockwise at w1. If I take in account the mass of the belt. The centrifugal forces are not the same at right than at left. So the support have a net torque on it. A motor on the support drive a the Pulley1. If the mass of the belt is 0, there is no torque on the support.



Fc1 > Fc2 so the support has a net torque on it. The mass of the belt must be >0 .



For cancel the torque on a pulley that turn counterclockwise I can use this method:



On the support I give energy to 2 pulleys, one clockwise, other counterclockwise. The clockwise pulley give a positive torque on the support because the motor don't give a torque. The counterclockwise pulley can give a torque to the motor from forces F, the motor need a torque. But this don't apply a torque on the support. There are a lot of small devices fixed on the belt, this devices can repuls themselves like magnet can do. It's possible to recover energy even it's not necessary because w1 can be very high if it receive a net torque from the clockwise pulley.




Or slow down the counterclockwise pulley like that:

neufneufneuf

2 motors, one drives clockwise other drives counterclockwise, like that there is no torque on the support from the motors. The clockwise pulley an give a clockwise torque on the device, I keep it. I try to cancel the energy of the counterclockwise disk without return a torque on the support. Purple disks can be placed in the center of the device (not like I drawn, but in the red point, in the center), the friction can be exist only on the center of the device is w4 is high. The angular velocity of purple disks can be very low relative to the support frame reference because there are in the center of the device. The energy needed by the green motor can be low IF the rotationnal velocity of purple disks is low relative to the support frame reference. There is friction between purple disks, like that the green torque can be canceled.

---

Maybe I can use a simple transmission (same diameter) with low quality belts, just only for let energy. If belt lost 10% of energy in friction, this energy can be transmit to the support.

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Or with this method, the motor is in the center (blue). The motor receives a clockwise torque. All torque on disks are canceled. There is friction between disks (red forces). It's possible to have a bigger diameter for the motor and 12 disks around the motor.




With a big disk and 2 other smaller. The diameter of the motor is very small, more than I drawn. If the diameter of the motor is near 0 and the bigger disk is very high, all the torque is canceled I think. The center of rotation of all the device is in the center of the green disk. Like that the device don't receive a torque from the green disk, and 2 small disk don't have a lot of torque.

neufneufneuf

neufneufneuf

On Earth:



It is like that, in the contrary I can create energy easily

So when the bubble is moving up the weight don't change I can use:



I don't decelerate with the bubble.

neufneufneuf

When I accelerate or decelerate something the center of gravity moves if I move something in it. So here, the object inside the container don't move (grey/white part). I return only the grey/wite part. I choose the density of the fluid =1. Inside the white part there is nothing, the density is 0 (theory). Inside the grey part it is a fluid of density 2. Like that the body to return has a global density =1. When I accelerate or decelerate the body (white/grey) don't move inside the blue fluid, the point A is always at the same position. When I return the grey/white part I recover energy because the grey part has a greater density than the white part. For return the grey/white part I don't decelerate, I just let the velocity constant and I return the grey/white part without need an energy. I can repeat the cycle. At final I can decelerate all the device and recover the energy I gave when I accelerated because the point A is always at the same position even I decelerate. I think it's possible to recover energy when the device is decelerating with return the grey/white part.




The shape of the grey/white part must be a cylinder, like that it's possible to rotate it around itself without energy.



Forces:



It's possible to use this idea in a circular device.


In fact, even the center of gravity moves, the force for accelerate the device is not in the CG it is where the pressure is, no ? And the pressure is always at the same point :



There is an axis of rotation (red). The axis receives the sum of forces, so the "weight" of the object is alsways like a full object with a density = 1

neufneufneuf

This question can be resumed: the force of pressure with a fluid is at the wall or in the center of mass ?

Look at this simple cycle:



I move up the container between the Earth and the Moon where gravity is 0. For me the force F exist at the wall not the center of mass. Even the center of mass moves the force is always at the wall. The energy needed for move up or move down the container is the same. The step3 don't need an energy because the gravity is 0. The step5 gives an energy. At final the device recover the energy at step5.

(imagine the Earth without an atmosphere, no friction no Archimede's principle from air).

neufneufneuf

Here the torque on the support is near 0. The ring and the disk are decelerating in the support frame reference. I recover the energy from friction and the ring and the disk increase their rotationnal velocity in the labo frame reference. I need only to adjust the velocity of the ring before to apply the belt.

Cycle:

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SET THE BELT OFF (imagine the device without the belt)

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1/ Turn the support at w, disk and ring don't turn in the labo frame reference. Disk and ring are turning counterclockwise -w in the support frame reference.
2/ The green disk must turn counterclockwise very quickly compare to the blue ring (frame reference) because diameters are not the same. I need to increase the velocity of the green disk counterclockwise (support frame reference), when I done that, the energy is conserved (for you it is always conserved) because the support receives a clockwise torque.

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SET THE BELT ON (now there is the belt)

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3/ I apply the force F1 to the blue ring from the red axis (I use a brake). F1 gives F2 to the support. The blue ring is decelerating, then the blue ring want to decelerate the green disk: this gives forces F5 and F6. These forces give F3 and F4 to the support. The support receives a clockwise torque from F3/F4. So the support don't receive a torque if the diameter of the green disk is very small.

The green disk is very small for cancel near all torque on the support but it can be attached to a bigger disk with a mass. The green disk I drawn is a pulley attached to a bigger disk. If the diameter of the blue ring is 0.1m and the diamter of the green disk is 0.001m, if w=1 rpm, then the angular velocity of the green disk must be at -100 rpm. If I apply a force F=20 N (a torque of 1Nm), the support receive a torque = 0.1/2*20 - (0.1-0.001)/2*20 = 0.01 Nm and it can be lower. The work on the green disk is 100Cwt=0.001/2*20*100 = 1*t. The work on the support is 0.01*w*t=0.01*t. So it's possible to compare these 2 different energies. The energy from the brake can be recover.

I think the blue ring must have a mass = 0 (or as lower as possible). Like that all the force F1 goes to the green disk. The green disk is decelerating in the support frame reference so it is accelerating in the labo frame reference. Even the diameter of the green disk is very small, it is attached to a bigger disk for have an inertia.

neufneufneuf

Like that:



The torque on the support is counterclockwise at -4/10dF, but the green ring receive 0.5dF and each blue pulley receives 0.1dF torque and at start w is the same angular velocity (even pulleys turn at -w)

Or with R2<R1 but near the same:



Pulleys can't have the same angular velocity, one must accelerate, other need to decelerate. Each pulley receives a torque FR, but the support receives the torque -F(R1-R2). And if I set the belt ON when the support is at w, 2 pulleys are turning at -w. So the work won by 2 pulleys is 2FRwt and the support lost F(R1-R2)wt.

Cycle:

1/ Turn the support with the belt OFF at w
2/ Set the belt ON
3/ The support receives -F(R1-R2), one disk FR1 the other disk FR2. FR1 increases kinetic energy and FR2 too.

If R1 is near R2 , angular velocities of 2 disks will be correct sortly after but during this moment the sum of energy of the device is not constant.

After the step 1/, the device is at w. But 2 disks don't turn in the labo frame reference, they don't have any kinetic energy around themselves. So if the angular velocity of 2 disks change, the bigger disk must decelerate (in the support frame reference) and the lower must accelerate (in the support frame reference). The bigger disk goes from for example -10 to -9 and the lower disk goes for example from -10 to -11. In the labo frame reference, 2 disks turns after the step3 so their kinetic energy are increasing.

The support gives -F(R1-R2) and one disk receives FR1 the other -FR2, but the torque on the smaller disk don't decrease its kinetic energy, because it don't turn in the labo frame reference, in the support frame reference each disk is turning at -w and -FR2 must decrease the angular velocity of the smaller disk, but not, the angular velocity is increasing.

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Compare these 2 cases:

No motor ! it's only a transcient analysis. All velocities are in labo frame reference.




Case 1/ At left, the support don't turn, the small disk turns at -6.2rd/s, the big disk turns at -2rd/s. The big disk will drive the smaller, I need a spring for adjust angular velocities. I noted R=R2. The big disk win 6wFRt, the small disk lost 6.2wF3Rt and the spring win the rest. The sum of energy won/lost is well 0.

Case 2/ At right, it's the same case except that the support is turning clockwise at w. The sum of forces are the same. Except one thing the support receive a clockwise torque +2wFRt. Here, the sum of energy won is 2wFRt.

Forces can be like I drawn because in the support frame reference: the bigger disk turns at -w and the smaller at -5.2w, the smaller turns more quickly even the radius is not the same.

In the second case the device win energy.

I could use the angular velocities (labo frame reference):

small disk: -3.6w
big disk: -0.5w
support: +w

In the support frame reference, the small disk turns at -4.6w and the bigger at -1.5w. So, the smaller disk attrack the bigger and forces can be like I drawn.

The sum of energy is:

-3.6wFRt+0.5w3RFt+2wFRt = -3wFRt

neufneufneuf

I can reverse the angular velocity without give a torque on the support if the diameter of P2 and P3 are very small. The motor gives -FRwt to the support but receives 2FRwt from pulleys. The motor needs the energy 2.1Frwt, P4 gives Frwt. P4 has a mass no others pulleys. The motor accelerates only P4. P2 drives P3 like a gear (not drawn). The sum is well at 0. So, I need to place at least 3 basic devices in serial.



And for reverse the velocity of bigger disk, I can use gears with small diameter in the center of each pulley P4 and P5, like that the torque on the support is near 0. The device can win 2FRwt

The power is transmit like that:

Motor is P1
P2 is driven from P1 with the belt
P3 is driven from P2 with the gear (each pulley has a gear fixed to it)
P4 is driven from P3 with the belt
P5 is driven from P4 with the gear inside P5 and inside P4 (small diameters)
P6 is driven from P5 with the belt
P7 is driven from P6 with the gear
P8 is driven from P7 with the belt

For resume:

neufneufneuf

With R = small radius. Big radius = 3R.

The support turns clockwise at w. There are 2 motors, they need -2*6FRwt and they give -2*3FRwt to the support. Pulleys of motors give +2*2FRwt. At the middle, green and purple pulleys give +4FRwt. Pulleys at right give +8FRwt. The disk at right receives +6FRwt. The sum is the torque from the green and purple pulleys in the middle give +4FRwt

I drawn 3 basic devices but there are more. With 3 the sum is at 0.

neufneufneuf

Or with a gyroscope like that. The goal is to decelerate (in the support frame reference) all purple disks because they turn at -w in the support frame reference. In the labo frame reference: disks don't turn (w=0), so if they decelerate in the support frame reference, they accelerate in the labo frame reference. The gyroscope don't turn, it can cancel near all torque from 2 ending disks. There is friction between disks and between the arm of the gyroscope and 2 ending disks. The gyroscope is on the support like disks.



Forces fx are more and more bigger with the angle, but it's possible to add a lot of gyros:


The gyroscope receives a torque around the axis z but it turns around the axis x.

I can place the disks in the 'center' like the gyroscope:

neufneufneuf

For reduce the angular velocity (in the support frame reference) I can use 2 same diameters sprockets with a roller chain. The roller chain lost energy in friction and gives a clockwise torque on the support. Imagine a roller chain with friction when I want to change the angle between of one link to another one. I can use a special roller chain for have forces like I want. The friction can exist only when it pass from straight to curve. Maybe there is no torque on the support but there is friction and sprocket decrease their angular velocity in the support frame reference.



I think forces are like that:


No torque on the support. But friction is an energy, and that energy must come from something when the support don't turn, and it's the kinetic energy of the sprocket.

neufneufneuf

The red/white container can accelerate very fast and give a torque to the support the work is FRwt. The weight in the container is always the same, this is the pressure at bottom that gives the 'weight' and even the red/white object rotates the weight don't change. The torque is applied on the support and the support turns at w so the work is FRwt. The red/white objet don't turn at start, it turns around itself but the angular velocity of the support don't add an angular velocity for the red/wite object.

Morbert

I think this thread has become too much of a monologue to remain open. Closing.

neufneufneuf, perhaps a website dedicated to Algodoo would be more appropriate.