DescentBB

How would I find the amount of energy it would take to move an object around in a circle at a constant speed v and mass m?

Is the speed angular or linear?

Energy = Force x Distance.

Force = Mass x Velocity.

Short answer: you can't find the answer unless you know the distance you want to move the object (hint: [spoiler]the circumference of the circle around which you're trying to move the object[/spoiler]).

Speed, by definition, has no direction whatsoever, as it's a scalar. You're probably confusing it with velocity. Angular velocity is a concept that only applies to objects that are spinning. Linear velocity refers to an object's motion through space.

Doing homework, I see?

actually, this isnt homework, just something im trying to figure out for fun

what I want to figure out is the amount of energy to rotate a mass around a circle of radius .1 meter with a linear velocity of .9c (its angular velocity would be 9c)

Actually, your velocity is linear, but your centripetal acceleration will be angular. I'm going to neglect friction and gravity because I don't have a Ph.D. in physics.

Force = Time Derivative of Momentum (m*v)

However, for an object where the mass does not change over time, this turns out to be:

Force = mass * acceleration

Centripetal acceleration is: v²/Radius

Therefore, the force exerted by the cord on the object is simply:

mass * v²/R

The kinetic energy of the mass can be determined at any time by:

K = .5 * m * v²

And I currently don't have any idea how to calculate how much energy it would take to get it up to that speed.

wouldnt the amount of energy required be the intergral of

K = .5 * m * v²

between the initial and final v

Yeah, that should work. And now... bed...

wouldnt the amount of energy required be the intergral of

K = .5 * m * v²

between the initial and final v

You said v was constant. Vi=Vf, so you can't intigrate with those as your limits.

"Force = Mass x Velocity"

booo

"Force = mass * acceleration "




Whats making it go in a circle? For something to go in a circle, it must be accelerating towards its center, whats making it do that?

If you are rolling it, then the shape of the mass comes into play. If you are like holding on to the end of a string and swinging around in a circle then that is something entirely different.

Also, DCrazy's energy equation is "work" energy. The other is Kinetic energy. We can also have potential.

It would be a rotating arm around a pivot, it would use the following propuplsion systems

a force being exerted from the end of the arm, perpindicular to the arm, like a jet or something

Correct me if I'm wrong (and I hope not, since I'm a physics major ), but if you're asking how much energy is required to move the object at constant velocity, the answer is none, at least without the presence of friction. Work is a force applied over a certain distance, and since your object isn't accelerating (at least not linearly), there's no force acting on it in that direction. Any object in circular motion has a tangential and radial component to its motion. The radial component gives you centripetal force, which changes the direction of the object's velocity but not its magnitude. Since you've already said that the linear velocity is constant, there's no force in that direction.

Now, if you were asking how much energy it would take to accelerate the object from rest to its final velocity, it would be a different story. By the work-kinetic energy theorem, the work performed on an object is equal to its change in kinetic energy. As far as I can remember from last year's class, since only the tangential force would contribute to the object's linear acceleration, you can treat the object in the same way that you would treat an object accelerating linearly. If that's the case, then the energy required to reach its final speed would simply equal its final kinetic energy. Note that I could very well be wrong in this case; I know that centripetal force keeps the object moving in a circle, but I'm not sure whether or not it would give any contribution to the actual change in its linear speed. My instincts tell me that it wouldn't, since it's perpendicular to the direction of the object's velocity, but I might be wrong.

Er, oops on the velocity/acceleration bit. Brain fart there.

TG, Energy would be required to move the object since it MUST move some distance dx in order to be considered "in motion". After getting it to its final velocity, however, no energy is required to maintain its velocity. The object has a constant acceleration due to centripetal force. It's tangential to the velocity, but it exists. It doesn't contribute to its speed (which isn't "linear" since it has no direction) but it constantly changes the directional component of the object's velocity vector. Learn to make that distinction in case you get a really stuck-up professor.

Plague, the acceleration is not angular. Angular acceleration is the d/dt of angular velocity.

By the way, a jet would provide a constant force -- constant acceleration, not a constant velocity.

And as a side note, TG, you shoulda made the short trip down 95 to Loyola Baltimore... Dane Cook performed here tonight. Tickets were $22.

i guess the main question is

is it possible to achieve FTL angular velocity without using infinite energy

vs

achieving FTL linear velocity with using more than infinite energy

Well, if everything is frictionless, and there is no gravity, once you get it going at speed v, it will keep going at speed v indefinitely. No energy required.

The energy you have to supply is the energy to get it moving from rest to speed v. Which would be force you supply times the distance required to make it go speed v. Then the force can be removed and it will go on forever.

Even with friction, once at speed V, the only energy required would be that which is needed to overcome the disapated energy through the friction of the joints.

Edit: Wow, alot of posts snuck in. I'd sign off on TG's post, and I have my BS in physics (took my GRE today). I havn't done these problems in 5 years though. Also, unfortionately I don't know what FTL is, and google just keep bringing back Fruit of the Loom (faster then light?)

Forgive me for inserting a tangental question here.
With ccb056's basic question, let us view a large wagon wheel structure in space. In fact let us view a very large wheel in the order of one light year in diameter.
Rotational speed is generated at the hub. Science tells us that FTL (faster than light) speed cannot be obtained. So, by slowly accelerating the hub speed up to say .95c wouldn't the rim speed of the wheel be going at faster than light? And what happens when the wheel is up to speed, as you traverse along a spoke from the hub to the rim? Have fun

Oh and lets forget about structural integrety of the wheel for the non eh.

Accelerating anything to or over the speed of light can be done inside the event horizon of a black hole. If you manage to get whatever you want to orbit the black hole inside the event horizon without being torn apart by the tidal forces gravity will be all you need to reach FTL velocity. I think I read or heard somewhere that any other method of accelerating something to over the speed of light would require infinite energy no matter how you tried to do it. So My guess to ccb's question is it would require 9 times infinity to accelerate to that velocity.

woodchip, you wouldn't be able to accelerate the center of the wheel any faster than it would take to get the outermost parts of the wheel to reach .99999... * c, because as velocity approaches c, mass approaches infinity. You'd need an infinitely increasing force (and therefore infinite energy) to just wind up approaching, but never reaching, c.

the linear velocity would be the same for every point on the wheel

the inear velocity would be less than the speed of light

I think that E=MC^2 only applies to linear velocity

It's the angular velocity that would be different at all point, and I dont think angular velocity follows the same mass and energy rules as linear velocity

the only way to prove this would be to find out how much energy it would take to get a body to rotate around a pivot with radius .1m at .9c

If the required energy is not infinite, then the angular velocity can in fact be faster than light because the only force that you are applying is in a linear direction, not angularly, right?

and, in a frictionless environment, would distance traveled really factor in when trying to determine the amount of energy to get a mass to reach a certain speed? because, if its just the area under the curve, it doesnt matter when you reach that speed, I think the area would be the same

also, keep in mind that the ratio of the linear velocity to the radius has to be greater than

299,792,458 to 1

and since the linear velocity cannot even equal 299,792,458 the only way to get around it is to have a radius less than 1 unit measure. ie .9c m/s to 1. m

It's all the same, you can not accelerate any part of the wheel to the speed of light. The laws apply to anything trying to move faster then light locally. The reason you can accelerate "faster then light" inside the event horizon of a spinning black hole is because the twisting "whirlpool" of space allows you to exceed the speed of light reletive to something outside the event horizon but still not locally.

Now if you really did start orbiting around in the event horizon of a black hole going FTL thats where the real fun starts since supposedly you would start going backwards in time. Or so they said on some TV show.

You don't need to worry about angular velocity; you can solve the problem in as follows:

Think about a typical spoke-and-wheel system.

1. Force = mass * acceleration.
2. Acceleration must be positive in order for velocity to increase towards c.
3. As an object's velocity approaches c, special relativity dictates that the object's mass approaches infinity.
4. As an object's mass approaches infinity, then by F=ma then the force exerted on the object must also approach infinity [lim(m -> +inf) ma = +inf].
5. As m approaches infinity, the ratio m/a approaches infinity as well, meaning that the faster the object is going, the more of that force is spent overcoming the object's inertia due to mass than actually accelerating the object.
6. Each "spoke" of the wheel can be thought of as accelerating a "slice" of the disk of mass dm.
7. If it's a disk of uniform density, then those pieces of mass can be integrated with respect to the angle about which you are rotating the disk (the integral from 0 to 2pi of dm with respect to theta).
8. Therefore, it is impossible to accelerate a disk to c about its internal axis.

I know that DCrazy, but you missed my point, you don't have to accelerate past C, you don't even have to accelerate to C, if you want, you can accelerate to .11c.

If your linear velocity on this wagon wheel is .11c, I don't think that you need an infinite amount of energy, right.

Now, lets take a point thats .1 units away from the center.

It's linear velocity is only .11c but it's angular velocity is 1.1c

but wait, we already established that you don't need infinite energy, but you're going 1.1c

ccb056 wrote:It's linear velocity is only .11c but it's angular velocity is 1.1c

No it's not. Angular velocity is in radians/sec [d(theta)/dt], c is defined in meters/sec (dx/dt).

right, but in this situation with radius .1m

if your angular velocity is 2pi radians per second
then
your angular velocity would also be .01pi meters per second

based on the relationship of
2pi radians = pi*.1^2 meters

You're completely forgetting about rotational inertia and angular momentum, both of which will yield the same infinite mass problem when you translate them into linear space.

radians can be used to obtain units

for example, when finding arc length, you need the radian measure of the angle to find the meter measure of the arc

what I am doing is using the same principle, but the arc is a full circle, and it is rotating at 1 cycle per second

as far as inertia and momentum, I would need formulas expressing that it would require either infinite mass or energy in order to beleive what you say is true

think of it like one of those booths at the carnival, where you shoot a paddle with a gun and it rotates

if you were to shoot the paddle with a pellet travelling at velocity .11c

and the distance from where it hit to the pivot was .1m

then the mass at impact would be the mass of the pellet + the mass of the paddle

from then on, no more mass or energy is added to the system

yet the paddle starts spinning

when its spinning, it's angular velocity is 1.1c

you added a finite amount of energy and/or mass

Well without going through the derivation of the cross product (the formula for angular momentum of mass m is defined as the vector cross-product of the radius and the linear momentum), the angular momentum L = mvr * sin(theta). As soon as you see that m in there you know there's a problem, because of the same reason shown above.

Regardless of where you apply the energy, it's still the same amount of energy. If things worked the way you were thinking everyone would push down on the short end of a lever.

The only way it would work is if the paddle has zero mass.

isn't the force around a circle's perimeter (whether inside, outside, circle or sphere) tangental?

I don't remember my formuli, but I do remember that an object moving in a circular direction is actually moving in an "infinate" amount of vectors that are perpendicular to the radius.

thus you get centripital force and not centrifical force.

Krom wrote:Regardless of where you apply the energy, it's still the same amount of energy. If things worked the way you were thinking everyone would push down on the short end of a lever.

right, torque

thats why it would work if you applied the force 1 unit away from the pivot, because the angular velocity would = the linear velocity

but if you apply the same force closer than 1 unit from the pivot the angular velocity is greater than the linear velocity

But you missed the most important part of Krom's post:

Krom wrote:The only way it would work is if the paddle has zero mass.

Otherwise, some of that energy is used to overcome the object's inertia.

not if the mass of the paddle = the mass of the pellet

or if the pellet had more mass than the paddle

You're confusing conservation of momentum with inertia. Plus at that point you have to consider impulse.

DCrazy wrote:TG, Energy would be required to move the object since it MUST move some distance dx in order to be considered "in motion". After getting it to its final velocity, however, no energy is required to maintain its velocity. The object has a constant acceleration due to centripetal force. It's tangential to the velocity, but it exists. It doesn't contribute to its speed (which isn't "linear" since it has no direction) but it constantly changes the directional component of the object's velocity vector. Learn to make that distinction in case you get a really stuck-up professor.

And as a side note, TG, you shoulda made the short trip down 95 to Loyola Baltimore... Dane Cook performed here tonight. Tickets were $22.

Yeah, I was intending to make the distinction between the energy required to accelerate the object and the lack of energy required to sustain its motion at constant velocity; I'm just not sure if I came across as saying so. My professor last year did make it a point to reinforce that centripetal acceleration is still acceleration even if the object's speed isn't changing; you're right in saying that it may seem trivial, but it's just the type of thing that can trip you up on an exam question. As for Dane Cook, I'm not a huge comedy fan, but his stuff is pretty good.

As for the rest of this thread, there are a lot of terms flying around that I'm positive that I learned last year, but I can't remember a single formula associated with them without consulting a textbook. Kind of depressing, really.

I can tell you now: no known material will stand the strain of rotating that bad boy at the velocity you want - not even if you counterbalance it with an identical mass on the opposite side of the circuit.

the material wouldn't have to be physical, it could be magnetic forces keeping everything together

Mobius wrote:I can tell you now: no known material will stand the strain of rotating that bad boy at the velocity you want - not even if you counterbalance it with an identical mass on the opposite side of the circuit.

Really? I would have never guessed that Professor OBVIOUS!

even if you did build a structurally sound contraption, no-one could survive the g-forces induced

it does have to be physical if you want to "hit" it with an object to study the effects. you can't "hit" a magnetic field.

but perhaps you are suggesting that the entire "paddle" be integrated with some sort of "structural integrity magnetic(?) field". sure why not, for the sake of the argument lets assume the paddle has infinite structural integrity along it's entire length, it cannot squash, stretch or break.

the paddle is traveling at different speeds along it's length. if the far end of the paddle is approaching C, then the "mass" of any respective section of paddle would be respectively exponentially* greater as you get towards the far end (the end approaching C).

*i assume this, as i don't know the equation for the relationship between mass and speed(relative to C).
[edit: lol it's E=MC^2, slaps forehead]

it wouldn't surprise me if the equations governing impulse answer ccb056's question like so: the moment the object strikes the paddle the paddle's mass becomes infinite (as it would theoretically be going faster than the speed of light), and it does not move at all.

but i'd like to know if this is correct or not, so i'll be watching this thread with interest .
such an interesting discussion

Would the mass become infinite if the linear velocity is less than the speed of light, no.

Would the mass change if the angular velocity changes, I don't think this has ever been adressed.

Angular vs Linear velocity, I don't think that the mass would become infinite because the driving thrust is linear, not angular.

The mass would only become infinite if the energy required would be infinite.

I don't think that infinite energy is required to spin a wheel.

What would be nice is to see all the equations related to this problem calcuated using the two following constants (Energy required to get linear velocity to .11c, impluse, momentum, G-forces, etc)

linear velocity =.11c (m/s)
radius = .1 (m)

As soon as that wheel hits a certain speed i think you'd be more worried about if it will hold together under all the inertial force.

ccb056 wrote:Would the mass become infinite if the linear velocity is less than the speed of light, no.

It will approach infinity as the speed approaches c. That limit is what matters.

ccb056 wrote:Would the mass change if the angular velocity changes, I don't think this has ever been adressed.

The mass of the different pieces will increase as they move towards the speed of light. Remember, internal motion can be translated to the linear world for a chunk of the object of mass dm, in which case the same rules apply.

ccb056 wrote:Angular vs Linear velocity, I don't think that the mass would become infinite because the driving thrust is linear, not angular.

Relativity knows no difference.

ccb056 wrote:The mass would only become infinite if the energy required would be infinite.

That statement violates causality. Required energy becomes infinite because mass becomes infinite, not vice versa.

ccb056 wrote:I don't think that infinite energy is required to spin a wheel.

It is in order to spin the wheel faster and faster towards c due to both rotational and linear inertia.