# Doubt in the length contraction of rod having velocity at some angle to its axis

I am reading AP French's Special Relativity, and I found an interesting problem there.

A rod of proper length L points along the $$x$$ axis but moves in a direction making an angle of $$45^°$$ to this axis (see the figure). A platform, also parallel to the $$x$$ axis, lies in the rod's way, but a slit of proper length $$1.1L$$ has been cut out of it so that the rod can easily fit through if it travels at a nonrelativistic speed. What happens if its speed is $$0.9c$$? Analyze the problem from both reference frames.

Ans- It fits through

As it is given that at non-relativistic speed , the rod fits into the slit. This means that length of rod=$$1.1L$$ (no length contraction). Also the centre of slit lies on the centre of rod when rod reaches the slit.

Consider a frame in which the rod appears to move with velocity $$0.9c$$ at $$45^o$$ to the $$x-$$ axis.
In its rest frame, the length of rod in the direction along the velocity is $$\frac{1.1L}{\sqrt{2}}$$ and perpendicular to the velocity is $$\frac{1.1L}{\sqrt{2}}$$. These are the rest lengths of the rod.
So the length of rod along the direction of velocity of rod becomes
$$l_{||}=\frac{1.1L}{\sqrt{2}}\sqrt{\Big(1-\frac{(0.9c)^2}{c^2}\Big)}=\frac{1.1L}{\sqrt{2}}\sqrt{(1-0.81)}=\frac{1.1L}{\sqrt{2}}\sqrt{0.19}<\frac{1.1L}{\sqrt{2}}$$

$$l_{\perp}=\frac{1.1L}{\sqrt{2}}$$

So from the second figure of the diagram given below, we can see that the dashed line is the $$l_{||}$$ which is less than actual length parallel to the velocity. $$l_{\perp}$$ remains the same. Thus the observer observes the tilting of the rod as shown by the red line.

Similarly in the frame in which rod appears to be at rest the slit becomes tilted.
But the answer is given that the rod will fit in the slit in both the frames.
How is it possible?

• Please show your work, this way it will be easier to notice your mistake. I don't have access to pen and paper, yet what I got in my head based on a few lines of computation agrees with the answer provided by the book. May 17, 2021 at 5:58
• I have included the diagram of what I think. You can see that from the diagram, which I have made. the rod is not horizontal when it is moving at speed $0.9c$ at $45^o$ with $x-$ axis. So how will it fit the slit? Because slit remains horizontal. That is the question?
– Iti
May 17, 2021 at 6:05
• Without showing how you got the length of the red line and etc. I think diagram itself isn't self explanatory. May 17, 2021 at 6:14
• I have updated the question showing my calculations. Please see it.
– Iti
May 17, 2021 at 6:29