# Rotating cylinder on x'-axis in S' frame. Find twist per unit length in S frame

• crissyb1988
In summary, when observing a cylinder rotating uniformly about the x' axis of S', which also travels forward, in the inertial frame of reference S, the twist per unit length is given by yωv/c(squared), where y is the gamma factor and ω is the angular speed. This can be proven using the Lorentz equations and the inverse Lorentz equations, taking into account the displacement of elements of the cylinder at different times in S'.
crissyb1988

## Homework Statement

A cylinder rotating uniformly about the x' axis of S' will seem twisted when observed instantaneously in S, where it not only rotates but also travels forward. If the angular speed of the cylinder in S' is ω, prove that in S the twist per unit length is yωv/c(squared). Here S and S' are inertial frames of reference in the standard configuration with respect to one another. y= gamma factor

## Homework Equations

twist per unit length = yωv/c(squared)
Lorentz equations
Inverse Lorentz equations

## The Attempt at a Solution

By twist per unit length ii think it means dθ/dx where the x-axis lines up with the axis of the cylinder?.

We can write the angular speed as
ω= dθ'/dt',
and then transposing we get
dθ'=ωdt'
because theta is in the z-y plane we can say that dθ'=dθ ?

So subbing in dt'=y(dt-vdx/c2) we get

dθ= ωy(dt-vdx/c2)

divide thru by dx we get

dθ/dx= ωy( dt/dx - v/c2)

dθ/dx = ωy/v - ωyv/c2

The answer should be dθ/dx = ωyv/c2 . BTW I don't think it matters about the negative sign but why am i left with ωy/v ?

Would really appreciate some hints :)

It's the comparison, at fixed time t in S of elements of the cylinder separated by dx AND that correspond to elements in S' with same θ' for a fixed t'. The thing is (x,t) and (x+dx,t) correspond to two different times t1' & t2' in S'. Elements of the cylinder with θ' at t1' are at θ'+ω(t2'-t1') at t2'.

## 1. What is a rotating cylinder on the x'-axis in the S' frame?

A rotating cylinder on the x'-axis in the S' frame is a physical system where a cylinder is rotating around its central axis, with the axis aligned with the x' axis in the S' frame of reference. This means that the cylinder is rotating in a specific direction and at a specific speed in relation to the S' frame of reference.

## 2. How is the S' frame related to the S frame?

The S' frame is a frame of reference that is moving at a constant velocity relative to the S frame. This means that the S' frame is a non-inertial frame, while the S frame is an inertial frame. The relationship between the two frames can be described using the Galilean transformation equations.

## 3. What is twist per unit length in the S frame?

Twist per unit length in the S frame refers to the amount of rotation per unit length that is observed in the S frame of reference. It is a measure of how much the cylinder is rotating in relation to the S frame, and can be calculated using the angular velocity of the cylinder and its length.

## 4. How is twist per unit length affected by the velocity of the S' frame?

Twist per unit length is affected by the velocity of the S' frame because the S' frame is a non-inertial frame. This means that the observed twist per unit length in the S frame will be different depending on the velocity of the S' frame. The faster the S' frame is moving, the greater the difference in observed twist per unit length will be.

## 5. What are some real-world applications of studying a rotating cylinder on the x'-axis in the S' frame?

Studying a rotating cylinder on the x'-axis in the S' frame can have various applications in different fields of science and engineering. For example, understanding the behavior of rotating cylinders can be useful in the design of engines, turbines, and other rotating machinery. It can also be applied in the study of fluid dynamics and the behavior of rotating fluids. Additionally, the principles involved in this system can be used in the development of gyroscopes and other navigation devices used in aircraft and spacecraft.

Replies
11
Views
2K
• Mechanics
Replies
1
Views
850
Replies
2
Views
2K
Replies
2
Views
2K
Replies
1
Views
2K
Replies
7
Views
1K
Replies
1
Views
5K
• Special and General Relativity
Replies
192
Views
16K
• Introductory Physics Homework Help
Replies
10
Views
2K
• General Engineering
Replies
1
Views
2K