# Speed greater than light in circular motion [duplicate]

We know that linear speed of object going around a circle is $\omega * r$ Now let us take an elastic string and rotate a body of negligible mass with $\omega = 500 rad/s$ It is possible to further stretch this string while maintaining $\omega$ constant using a super powerful motor.

If we extend the chord length to say $1,000,000 m$ then the linear speed of the body should come out to be equal to $500,000,000 m/s$ which is greater than the speed of the light. Where is the fallacy in the above argument?

• What mass do you mean by "negligible"? Special relativity allows the mass even be exactly zero. Such a mass will always travel at $c$ though. – Ruslan Aug 22 '14 at 6:19
• Behold to the speed of light right value. – Py-ser Aug 22 '14 at 6:35
• @Py-Ser Thanks! I have corrected the values – bhavesh Aug 22 '14 at 6:54
• Related: physics.stackexchange.com/q/8659/2451 and links therein. – Qmechanic Aug 22 '14 at 8:13

Let's have a look at your setup for spinning the mass at the end of the cord: To accelerate the mass you need to be able to apply a tangential force, and this requires the angle between the cord and the line to the pivot point be greater than zero. Specifically, if the tension in the cord is $T$ then the tangential acceleration is:

$$a_t = T\sin\theta$$

To achieve this either the pivot being driven by your motor needs a finite radius (as shown in the diagram) or the point where the string is attached needs to be moved in a circle (usually the case if you're spinning something around with your finger).

Either way, the force is transmitted to the mass through the cord. As the velocity of the mass approaches $c$ the tension in the cord rises to $\infty$. Unless you have an infinitely strong cord, before the mass can reach $c$ the cord will snap. Even if you had an infinitely strong cord you'd need an infinite amount of energy for the mass to reach $c$.

I think the basic misunderstanding is that you are using classical mechanics in a situation that needs relativistic mechanics.

Even if you have a strong chord relativistic mechanics tells us that as the linear velocity approaches c the inertial mass becomes infinite. That the linear velocity is increased tangentially is an irrelevant detail .

Note that what you describe is:

applying enough pressure, to make an object go very fast.

It makes absolutely no difference if your technique for doing that is "A BMW engine" "A Ferrari engine" or "swinging on string" -- or, any other technique!

Your question is, essentially, the same as if you asked: "I know you can't make something go faster than the speed of light using a BMW engine, but what if I used a Ferrari engine..."

Your technique for applying force to the object (engines, rockets, string, spinning ... whatever) makes no difference!

Hopefully this is a simple way to see the situation here.