How long must the second hand of a watch be for the tip to be traveling faster than the speed of light?

Question

How long must the second hand of a watch be for the tip to be traveling faster than the speed of light?

Answer 1

Let c = speed of light, d = length of second hand.

Second hand moves a t angular speed, w = 2*pi/60 radians /second (It make one full revolutionin one minute or 60 seconds). So w is about 0.1 rad/s

Now linear speed is equal to angluar speed time the radius about which the motion is taking place:

v = w * r or for teh second hand

c = w * d ---> d = c/w =3x10^8/0.1 = 3x10^9 meters

So the second hand has to be longer than 3 billion meters

Answer 2

Relativistically, it has to be infinitely long. Without doing the math, I just know weird things happen to a second hand at that speed. It will warp and spiral in some way, probably.

Classically, it's just a question of doing the math. Call the length of the second hand R. It takes one minute for the tip to complete a circuit of 2*pi*R, so the velocity is 2*pi*R / 60 seconds. Solve the equation 2*pi*R/60 = c for R, and you're done.

Answer 3

We can use the distance/time = velocity formula

distance = 2πR
time = 60 seconds
velocity = speed of light = 186,282 miles/hr = 186,282 miles/3600 sec

putting all of this together we get:

2πR/60 = 186,282/3600

R = (186,282) (60) / (3600) (2) (3.14)

R = 494.378 miles = 494.378 x 5280 = 2,610,321 feet

Wow, nice problem.

Ofcourse, considering relativity, the mass of the second hand would become infinite, while its thickness becomes zero. Also time itself would slow to zero. Luckily, we are considering a mathematical world.

Answer 4

1 meter